Abstract

An alternative method to compute Seidel aberrations is presented that utilizes real pseudoparaxial skew rays traveling through a rotationally-symmetric lens system as a Taylor series expansion in terms of object height and ray-direction spherical coordinates. Expressions for defocus, lateral magnification, and Seidel primary ray aberration coefficients are obtained in terms of numerically-determined higher-order partial derivatives of rays which are proximate to the optical axis. In contrast to commonly used methods, the new form of the aberration coefficients is related to unit entrance pupil radius and unit object height. The proposed methodology can be extended to derive higher-order ray aberration coefficients.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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  1. W. J. Smith, Modern Optical Engineering, 3rd ed. (McGraw-Hill, 2000) p. 63.
  2. R. B. Johnson, “A historical perspective on understanding optical aberrations,” Proc. SPIE 10263, 1026303 (1992).
    [Crossref]
  3. H. A. Buchdahl, Optical Aberration Coefficients (Dover, 1968).
  4. W. T. Welford, Aberrations of Optical Systems (Adam Hilger, 1986).
  5. G. W. Hopkins, “Proximate ray tracing and optical aberration coefficients,” J. Opt. Soc. Am. 66(5), 405–410 (1976).
    [Crossref]
  6. W. T. Welford, “A new total aberration formula,” J. Mod. Opt. 19, 719–727 (1972).
    [Crossref]
  7. B. Chen and A. M. Herkommer, “High order surface aberrations contributions from phase space analysis of differential rays,” Opt. Express 24(6), 5934–5945 (2016).
    [Crossref]
  8. M. Oleszko, R. Hambach, and H. Gros, “Decomposition of the total wave aberration in generalized optical systems,” J. Opt. Soc. Am. 34(10), 1856–1864 (2017).
    [Crossref]
  9. F. Bociort and J. Kross, “Seidel aberration coefficients for radial gradient-index lenses,” J. Opt. Soc. Am. A 11(10), 2647–2656 (1994).
    [Crossref]
  10. D. Claus, J. Watson, and J. Rodenburg, “Analysis and interpretation of the Seidel aberration coefficients in digital holography,” Appl. Opt. 50(34), H220–H229 (2011).
    [Crossref]
  11. R. S. Chang, J. Y. Sheu, and C. H. Lin, “Analysis of Seidel aberration by use of the discrete wavelet transform,” Appl. Opt. 41(13), 2408–2413 (2002).
    [Crossref]
  12. R. Kingslake and R. B. Johnson, Lens Design Fundamentals, 2 Edition (Academic, 2010).
  13. ibid. Sec. 4.4.
  14. ibid. Sec. 3.1.8.
  15. J. Sasián, Introduction to Aberrations in Optical Imaging Systems (Cambridge, 2013).
  16. R. B. Johnson, “Polynomial ray aberrations computed in various lens design programs,” Appl. Opt. 12(9), 2079–2082 (1973).
    [Crossref]
  17. R. K. Tyson, “Conversion of Zernike aberration coefficients to Seidel and higher-order power-series aberration coefficients,” Opt. Lett. 7(6), 262–264 (1982).
    [Crossref]
  18. G. Conforti, “Zernike aberration coefficients from Seidel and higher-order power-series coefficients,” Opt. Lett. 8(7), 407–408 (1983).
    [Crossref]
  19. R. B. Johnson, “Image defects useful in teaching students,” Proc. SPIE 0766, 10–17 (1987).
    [Crossref]
  20. R. B. Johnson, “Balancing the astigmatic fields when all other aberrations are absent,” Appl. Opt. 32(19), 3494–3496 (1993).
    [Crossref]
  21. P. D. Lin, Advanced Geometrical Optics (Springer, 2016), Eq. (7.30).
  22. ibid. Chap. 6.
  23. ibid. Eq. (2.37).
  24. The units of length and angle used in example are millimeters and radians, respectively.
  25. “Zemax OpticStudio 18.9 User Manual,” (Zemax LLC, 2018). p. 970.
  26. ibid. FIFTHORD macro, p. 2080.
  27. In the Zemax® optical design and analysis program, the signs of both the unconverted and converted Seidel coefficients are reversed; however, the FIFTHORD macro by Rimmer included in Zemax® does have the correct signs.
  28. A. E. Conrady, Applied Optics and Optical Design (Dover, 1957).
  29. M. Herzberger, Modern Geometrical Optics (Interscience, 1958).
  30. A. Cox, A System of Optical Design (Focal, 1964).
  31. W. Brouwer, Matrix Methods on Optical Instrument Design (Benjamin, 1964).
  32. R. K. Luneburg, Mathematical Theory of Optics (University of California, 1966).
  33. G. G. Slyusarev, Aberration and Optical Design Theory (Adam Hilger, 1984).

2017 (1)

M. Oleszko, R. Hambach, and H. Gros, “Decomposition of the total wave aberration in generalized optical systems,” J. Opt. Soc. Am. 34(10), 1856–1864 (2017).
[Crossref]

2016 (1)

2011 (1)

2002 (1)

1994 (1)

1993 (1)

1992 (1)

R. B. Johnson, “A historical perspective on understanding optical aberrations,” Proc. SPIE 10263, 1026303 (1992).
[Crossref]

1987 (1)

R. B. Johnson, “Image defects useful in teaching students,” Proc. SPIE 0766, 10–17 (1987).
[Crossref]

1983 (1)

1982 (1)

1976 (1)

1973 (1)

1972 (1)

W. T. Welford, “A new total aberration formula,” J. Mod. Opt. 19, 719–727 (1972).
[Crossref]

Bociort, F.

Brouwer, W.

W. Brouwer, Matrix Methods on Optical Instrument Design (Benjamin, 1964).

Buchdahl, H. A.

H. A. Buchdahl, Optical Aberration Coefficients (Dover, 1968).

Chang, R. S.

Chen, B.

Claus, D.

Conforti, G.

Conrady, A. E.

A. E. Conrady, Applied Optics and Optical Design (Dover, 1957).

Cox, A.

A. Cox, A System of Optical Design (Focal, 1964).

Gros, H.

M. Oleszko, R. Hambach, and H. Gros, “Decomposition of the total wave aberration in generalized optical systems,” J. Opt. Soc. Am. 34(10), 1856–1864 (2017).
[Crossref]

Hambach, R.

M. Oleszko, R. Hambach, and H. Gros, “Decomposition of the total wave aberration in generalized optical systems,” J. Opt. Soc. Am. 34(10), 1856–1864 (2017).
[Crossref]

Herkommer, A. M.

Herzberger, M.

M. Herzberger, Modern Geometrical Optics (Interscience, 1958).

Hopkins, G. W.

Johnson, R. B.

R. B. Johnson, “Balancing the astigmatic fields when all other aberrations are absent,” Appl. Opt. 32(19), 3494–3496 (1993).
[Crossref]

R. B. Johnson, “A historical perspective on understanding optical aberrations,” Proc. SPIE 10263, 1026303 (1992).
[Crossref]

R. B. Johnson, “Image defects useful in teaching students,” Proc. SPIE 0766, 10–17 (1987).
[Crossref]

R. B. Johnson, “Polynomial ray aberrations computed in various lens design programs,” Appl. Opt. 12(9), 2079–2082 (1973).
[Crossref]

R. Kingslake and R. B. Johnson, Lens Design Fundamentals, 2 Edition (Academic, 2010).

Kingslake, R.

R. Kingslake and R. B. Johnson, Lens Design Fundamentals, 2 Edition (Academic, 2010).

Kross, J.

Lin, C. H.

Lin, P. D.

P. D. Lin, Advanced Geometrical Optics (Springer, 2016), Eq. (7.30).

Luneburg, R. K.

R. K. Luneburg, Mathematical Theory of Optics (University of California, 1966).

Oleszko, M.

M. Oleszko, R. Hambach, and H. Gros, “Decomposition of the total wave aberration in generalized optical systems,” J. Opt. Soc. Am. 34(10), 1856–1864 (2017).
[Crossref]

Rodenburg, J.

Sasián, J.

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

Sheu, J. Y.

Slyusarev, G. G.

G. G. Slyusarev, Aberration and Optical Design Theory (Adam Hilger, 1984).

Smith, W. J.

W. J. Smith, Modern Optical Engineering, 3rd ed. (McGraw-Hill, 2000) p. 63.

Tyson, R. K.

Watson, J.

Welford, W. T.

W. T. Welford, “A new total aberration formula,” J. Mod. Opt. 19, 719–727 (1972).
[Crossref]

W. T. Welford, Aberrations of Optical Systems (Adam Hilger, 1986).

Appl. Opt. (4)

J. Mod. Opt. (1)

W. T. Welford, “A new total aberration formula,” J. Mod. Opt. 19, 719–727 (1972).
[Crossref]

J. Opt. Soc. Am. (2)

M. Oleszko, R. Hambach, and H. Gros, “Decomposition of the total wave aberration in generalized optical systems,” J. Opt. Soc. Am. 34(10), 1856–1864 (2017).
[Crossref]

G. W. Hopkins, “Proximate ray tracing and optical aberration coefficients,” J. Opt. Soc. Am. 66(5), 405–410 (1976).
[Crossref]

J. Opt. Soc. Am. A (1)

Opt. Express (1)

Opt. Lett. (2)

Proc. SPIE (2)

R. B. Johnson, “Image defects useful in teaching students,” Proc. SPIE 0766, 10–17 (1987).
[Crossref]

R. B. Johnson, “A historical perspective on understanding optical aberrations,” Proc. SPIE 10263, 1026303 (1992).
[Crossref]

Other (20)

H. A. Buchdahl, Optical Aberration Coefficients (Dover, 1968).

W. T. Welford, Aberrations of Optical Systems (Adam Hilger, 1986).

W. J. Smith, Modern Optical Engineering, 3rd ed. (McGraw-Hill, 2000) p. 63.

R. Kingslake and R. B. Johnson, Lens Design Fundamentals, 2 Edition (Academic, 2010).

ibid. Sec. 4.4.

ibid. Sec. 3.1.8.

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

P. D. Lin, Advanced Geometrical Optics (Springer, 2016), Eq. (7.30).

ibid. Chap. 6.

ibid. Eq. (2.37).

The units of length and angle used in example are millimeters and radians, respectively.

“Zemax OpticStudio 18.9 User Manual,” (Zemax LLC, 2018). p. 970.

ibid. FIFTHORD macro, p. 2080.

In the Zemax® optical design and analysis program, the signs of both the unconverted and converted Seidel coefficients are reversed; however, the FIFTHORD macro by Rimmer included in Zemax® does have the correct signs.

A. E. Conrady, Applied Optics and Optical Design (Dover, 1957).

M. Herzberger, Modern Geometrical Optics (Interscience, 1958).

A. Cox, A System of Optical Design (Focal, 1964).

W. Brouwer, Matrix Methods on Optical Instrument Design (Benjamin, 1964).

R. K. Luneburg, Mathematical Theory of Optics (University of California, 1966).

G. G. Slyusarev, Aberration and Optical Design Theory (Adam Hilger, 1984).

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Figures (3)

Fig. 1.
Fig. 1. Ray originating from an object, passing through entrance pupil, and then intersecting the image plane. The ${\textrm{x}_0}\;{\textrm{and}\ }{\textrm{y}_0}$ axes are shown displaced from the object plane for clarity.
Fig. 2.
Fig. 2. Entrance pupil with Cartesian coordinates $({\textrm{x}_\textrm{a}},{\textrm{y}_\textrm{a}})$ and polar coordinates $(\rho ,\phi )$.
Fig. 3.
Fig. 3. Example rotationally-symmetric optical system having a Tessar type configuration.

Tables (4)

Tables Icon

Table 1. Prescription for example rotationally-symmetric optical system shown in Fig. 3. The entrance pupil is located 22.103275 from the vertex of surface 1.

Tables Icon

Table 2. Third-order derivatives computed by three Finite-Difference (FD) methods.

Tables Icon

Table 3. The percentage errors, relative to the values in Table 2, of the third-order derivatives estimated by different FD methods.

Tables Icon

Table 4. Terms necessary to compute B3 and B4.

Equations (60)

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P ¯ n = [ P nx P ny P nz 1 ] T = [ Δ P nx Δ P ny P nz 1 ] T
Δ P nx = A 1 ( ρ S ϕ ) + B 1 ( ρ 3 S ϕ ) + B 2 h 0 ρ 2 S ( 2 ϕ ) + ( B 3 + B 4 ) h 0 2 ( ρ S ϕ ) and
Δ P ny = A 2 h 0 + [ A 1 ( ρ C ϕ ) + B 1 ( ρ 3 C ϕ ) + B 2 h 0 ρ 2 [ 2 + C ( 2 ϕ ) ] + ( 3 B 3 + B 4 ) h 0 2 ( ρ C ϕ ) + B 5 h 0 3 ] .
P ¯ 0 = [ P 0 x P 0 y P 0 z 1 ] T = [ 0 h 0 P 0 z 1 ] T
¯ 0 = [ 0 x 0 y 0 z 0 ] T = [ S α 0 C β 0 S β 0 C α 0 C β 0 0 ] T .
X ¯ 0 = [ 0 h 0 P 0 z α 0 β 0 ] T .
X ¯ aber = [ h 0 α 0 β 0 ] T .
P ¯ n ( X ¯ aber ) = [ Δ P nx Δ P ny P nz 1 ] T
Δ P nx = ( P nx X ¯ aber ) X ¯ aber + 1 2 X ¯ aber T ( 2 P nx X ¯ aber 2 ) X ¯ aber + 1 6 X ¯ aber T ( X ¯ aber T 3 P nx X ¯ aber 3 ) X ¯ aber + Δ P nx / Jacobian + Δ P nx/Hessian + Δ P nx/third + and
Δ P ny = ( P ny X ¯ aber ) X ¯ aber + 1 2 X ¯ aber T ( 2 P ny X ¯ aber 2 ) X ¯ aber + 1 6 X ¯ aber T ( X ¯ aber T 3 P ny X ¯ aber 3 ) X ¯ aber + Δ P ny/Jacobian + Δ P ny/Hessian + Δ P ny/third +
X ¯ aber/optical axis = [ 0 0 0 ] T .
Δ P nx/Jacobian = P nx h 0 h 0 + P nx α 0 α 0 + P nx β 0 β 0 ,
Δ P ny/Jacobian = P ny h 0 h 0 + P ny α 0 α 0 + P ny β 0 β 0 ,
Δ P nx/Hessian = 1 2 ( 2 P nx h 0 2 h 0 2 + 2 P nx α 0 2 α 0 2 + 2 P nx β 0 2 β 0 2 + 2 2 P nx h 0 α 0 h 0 α 0 + 2 2 P nx h 0 β 0 h 0 β 0 + 2 2 P nx α 0 β 0 α 0 β 0 ) ,
Δ P ny/Hessian = 1 2 ( 2 P ny h 0 2 h 0 2 + 2 P ny α 0 2 α 0 2 + 2 P ny β 0 2 β 0 2 + 2 2 P ny h 0 α 0 h 0 α 0 + 2 2 P ny h 0 β 0 h 0 β 0 + 2 2 P ny α 0 β 0 α 0 β 0 ) ,
Δ P nx/third = 1 6 [ 3 P nx α 0 3 α 0 3 + 3 P nx β 0 3 β 0 3 + 3 P nx h 0 3 h 0 3 + 6 3 P nx h 0 α 0 β 0 h 0 α 0 β 0 + 3 ( 3 P nx α 0 β 0 2 α 0 β 0 2 + 3 P nx α 0 2 β 0 α 0 2 β 0 + 3 P nx h 0 α 0 2 h 0 α 0 2 + 3 P nx h 0 β 0 2 h 0 β 0 2 + 3 P nx h 0 2 α 0 h 0 2 α 0 + 3 P nx h 0 2 β 0 h 0 2 β 0 ) ] ,
Δ P ny/third = 1 6 [ 3 P ny α 0 3 α 0 3 + 3 P ny β 0 3 β 0 3 + 3 P ny h 0 3 h 0 3 + 6 3 P ny h 0 α 0 β 0 h 0 α 0 β 0 + 3 ( 3 P ny α 0 β 0 2 α 0 β 0 2 + 3 P ny α 0 2 β 0 α 0 2 β 0 + 3 P ny h 0 α 0 2 h 0 α 0 2 + 3 P ny h 0 β 0 2 h 0 β 0 2 + 3 P ny h 0 2 α 0 h 0 2 α 0 + 3 P ny h 0 2 β 0 h 0 2 β 0 ) ] .
Δ P nx = P nx α 0 α 0 + 1 6 3 P nx α 0 3 α 0 3 + 1 2 3 P nx α 0 β 0 2 α 0 β 0 2 + 3 P nx h 0 α 0 β 0 h 0 α 0 β 0 + 1 2 3 P nx h 0 2 α 0 h 0 2 α 0 and
Δ P ny = P ny h 0 h 0 + P ny β 0 β 0 + 1 6 3 P ny β 0 3 β 0 3 + 1 2 3 P ny α 0 2 β 0 α 0 2 β 0 + 1 2 3 P ny h 0 α 0 2 h 0 α 0 2 + 1 2 3 P ny h 0 β 0 2 h 0 β 0 2 + 1 2 3 P ny h 0 2 β 0 h 0 2 β 0 + 1 6 3 P ny h 0 3 h 0 3
P ¯ n X ¯ aber = [ P nx / P nx h 0 h 0 P nx / P nx α 0 α 0 P nx / P nx β 0 β 0 P ny / P ny h 0 h 0 P ny / P ny α 0 α 0 P ny / P ny β 0 β 0 P nz / P nz h 0 h 0 P nz / P nz α 0 α 0 P nz / P nz β 0 β 0 ]
2 P ¯ n X ¯ aber 2 = [ 2 P nx h 0 2 2 P nx h 0 α 0 2 P nx h 0 β 0 2 P nx α 0 2 2 P nx α 0 β 0 2 P nx β 0 2 2 P ny h 0 2 2 P ny h 0 α 0 2 P ny h 0 β 0 2 P ny α 0 2 2 P ny α 0 β 0 2 P ny β 0 2 2 P nz h 0 2 2 P nz h 0 α 0 2 P nz h 0 β 0 2 P nz α 0 2 2 P nz α 0 β 0 2 P nz β 0 2 ]
3 P nx h 0 α 0 β 0 = ( 2 P nx α 0 β 0 ) 2 ( 2 P nx α 0 β 0 ) 1 Δ h 0 .
3 P nx h 0 α 0 β 0 = ( 2 P nx h 0 β 0 ) 2 ( 2 P nx h 0 β 0 ) 1 Δ α 0
3 P nx h 0 α 0 β 0 = ( 2 P nx h 0 α 0 ) 2 ( 2 P nx h 0 α 0 ) 1 Δ β 0 .
[ x a y a z a 1 ] = [ C ϕ S ϕ 0 0   S ϕ C ϕ 0 0 0 0 1 0 0 0 0 1 ] [ 0 ρ 0 1 ] = [ ρ S ϕ ρ C ϕ 0 1 ] .
x a = ρ S ϕ and
y a = ρ C ϕ .
Δ P nx = A 1 x a + B 1 ( x a 2 + y a 2 ) x a + 2 B 2 h 0 x a y a + ( B 3 + B 4 ) h 0 2 x a and
Δ P ny = A 2 h 0 + [ A 1 y a + B 1 ( x a 2 + y a 2 ) y a + B 2 h 0 ( x a 2 + 3 y a 2 ) + ( 3 B 3 + B 4 ) h 0 2 y a + B 5 h 0 3 ] .
Δ P nx = P nx α 0 α 0 + 1 6 3 P nx α 0 3 α 0 3 + 1 2 3 P nx α 0 β 0 2 ( β 0 β 0 / prin ) 2 α 0 + 3 P nx h 0 α 0 β 0 h 0 α 0 ( β 0 β 0 / prin ) + [ { 3 P nx h 0 α 0 β 0 + 3 P nx α 0 β 0 2 ( β 0 h 0 ) } ( β 0 / prin h 0 ) 1 2 { 3 P nx α 0 β 0 2 ( β 0 / prin h 0 ) 2 3 P nx h 0 2 α 0 } ] h 0 2 α 0
Δ P ny = P ny β 0 β 0 + 1 6 3 P ny β 0 3 ( β 0 β 0 / prin ) 3 + 1 2 3 P ny α 0 2 β 0 α 0 2 ( β 0 β 0 / prin ) + 1 2 [ 2 ( P ny h 0 ) prin + 3 P ny h 0 α 0 2 α 0 2 + 3 P ny h 0 β 0 2 ( β 0 β 0 / prin ) 2 ] h 0 + [ 1 2 3 P ny α 0 2 β 0 ( α 0 h 0 ) 2 β 0 / prin ( β 0 β 0 / prin ) + 3 P ny h 0 β 0 2 ( β 0 / prin h 0 ) + 1 2 3 P ny β 0 3 ( β 0 h 0 ) ( β 0 / prin h 0 ) + 1 2 3 P ny h 0 2 β 0 ] h 0 2 ( β 0 β 0 / prin ) + [ 1 6 3 P ny β 0 3 ( β 0 / prin h 0 ) 3 + 1 2 3 P ny h 0 β 0 2 ( β 0 / prin h 0 ) 2 + 1 2 3 P ny h 0 2 β 0 ( β 0 / prin h 0 ) + 1 6 3 P ny h 0 3 ] h 0 3 .
A 1 = P nx α 0 α 0 x a = P ny β 0 β 0 y a ,
A 2 = ( P ny h 0 ) prin ,
B 1 = ( 1 6 3 P nx α 0 3 α 0 3 + 1 2 3 P nx α 0 β 0 2 ( β 0 β 0 / prin ) α 0 2 ) / ( 1 6 3 P nx α 0 3 α 0 3 + 1 2 3 P nx α 0 β 0 2 ( β 0 β 0 / prin ) α 0 2 ) [ ( x a 2 + y a 2 ) y a ] [ ( x a 2 + y a 2 ) y a ] ,
B 1 = ( 1 6 3 P ny β 0 3 ( β 0 β 0 / prin ) 3 + 1 2 3 P ny α 0 2 β 0 ( β 0 β 0 / prin ) α 0 2 ) / ( 1 6 3 P ny β 0 3 ( β 0 β 0 / prin ) 3 + 1 2 3 P ny α 0 2 β 0 ( β 0 β 0 / prin ) α 0 2 ) [ ( x a 2 + y a 2 ) y a ] [ ( x a 2 + y a 2 ) y a ] ,
B 2 = 1 2 ( 3 P nx h 0 α 0 β 0 ) prin ( α 0 x a ) ( β 0 β 0 / prin y a ) ,
B 2 = 1 2 ( 3 P ny h 0 α 0 2 ) prin α 0 2 ( x a 2 + 3 y a 2 ) + 1 2 ( 3 P ny h 0 β 0 2 ) prin ( β 0 β 0 / prin ) 2 ( x a 2 + 3 y a 2 ) ,
B 3 + B 4 = { [ 3 P nx h 0 α 0 β 0 + 3 P nx α 0 β 0 2 ( β 0 h 0 ) ] ( β 0 / prin h 0 ) + 1 2 [ 3 P nx h 0 2 α 0 3 P nx α 0 β 0 2 ( β 0 / prin h 0 ) 2 ] } ( α 0 x a )
3 B 3 + B 4 = { 1 2 3 P ny h 0 2 β 0 + [ 3 P ny h 0 β 0 2 + 1 2 3 P ny β 0 3 ( β 0 h 0 ) ] ( β 0 / prin h 0 ) } ( β 0 y a )
B 5 = 1 6 3 P ny β 0 3 ( β 0 / prin h 0 ) 3 + 1 2 3 P ny h 0 β 0 2 ( β 0 / prin h 0 ) 2 + 1 2 3 P ny h 0 2 β 0 ( β 0 / prin h 0 ) + 1 6 3 P ny h 0 3 .
[ x a y a ] = [ 0 h 0 ] + ( 200 + d entrance ) C α 0 C β 0 [ S α 0 C β 0 S β 0 ] .
α 0 / prin = 0
β 0 / prin = tan 1 ( h 0 200 + d entrance ) .
B 1 = 1 6 3 P ny β 0 3 ( β 0 y a ) 3 = 1.65550609 × 10 4 ,
β 0 y a = ( 200 + d entrance ) C α 0 ( 200 + d entrance ) 2 + [ ( y a h 0 ) C α 0 ] 2 = 4.50240997 × 10 3 .
( 3 P nx h 0 α 0 β 0 ) prin = 1.48465920
( 3 P ny h 0 β 0 2 ) prin = 4.45397784.
α 0 x a = α 0 x a = ( 200 + d entrance ) ( 200 + d entrance ) 2 + x a 2 = 4.50240997 × 10 3
β 0 / prin y a = β 0 / prin y a = 0.
B 2 = 1 2 ( 3 P nx h 0 α 0 β 0 ) prin ( α 0 x a ) ( β 0 β 0 / prin y a ) = 1.50482796 × 10 5 .
α 0 = x a = 0
B 2 = 1 6 ( 3 P ny h 0 β 0 2 ) prin ( β 0 y a β 0 / prin y a ) 2 = 1.50482804 × 10 5
B 3 + B 4 = 4.19415619 × 10 6
3 B 3 + B 4 = 1.50363962 × 10 6 .
B 3 = ( 1 4 A 2 ) [ 3 P ny h 0 2 β 0 ( β 0 y a ) 2 3 P nx h 0 2 α 0 ( α 0 x a ) 2 . ]
B 4 = 3 4 3 P nx h 0 2 α 0 ( α 0 x a ) 1 4 3 P ny h 0 2 β 0 ( β 0 y a . )
3 P ny β 0 3 = 1.08829572 × 10 4 ,
3 P ny h 0 β 0 2 = 53.4535130 ,
3 P ny h 0 2 β 0 = 0.261391190 and
3 P ny h 0 3 = 1.28704975 × 10 3 .

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