Abstract

Conventional paraxial ray-tracing procedures are widely used for optical systems design and analysis. However, they are not applicable to multiple-dispersion-prism systems. Accordingly, the present study simplifies the equations given by the present group in a previous paper [Optik 117, 329 (2006) [CrossRef]  ] to the form of 3×3 matrix equations for tracing paraxial rays in optical systems containing triangular prisms. The accuracy and validity of the proposed approach are demonstrated by means of four numerical examples. The results confirm that the proposed equations provide a convenient and practical tool for analyzing paraxial rays traveling through non-axially symmetrical optical systems containing triangular and rectangular prisms.

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References

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  1. J. P. Rolland, “The art of back-of-the-envelope paraxial raytracing,” IEEE Trans. Educ. 44, 365–372 (2001).
    [Crossref]
  2. A. Gerrard and J. M. Burch, Introduction to Matrix Methods in Optics (Wiley, 1975).
  3. M. H. Freeman, Optics, 10th ed. (Butterworths, 1990).
  4. R. Guenther, “Geometrical optics,” in Modern Optics (Wiley, 1990), pp. 129–212.
  5. D. S. Goodman, “General principles of geometrical optics,” in Handbook of Optics, M. Bass, ed. (McGraw-Hill, 1995), Vol. 1, Chap. 1.
  6. F. L. Pedrotti and S. L. Pedrotti, Introduction to Optics, 2nd ed. (Prentice-Hall, 1996).
  7. E. Hecht, Optics, 3rd ed. (Addison-Wesley, 1998).
  8. D. J. Won, “Propagation of a general Gaussian beam by paraxial ray tracing in an arbitrary optical system,” J. Korean Phys. Soc. 58, 735–741 (2011).
    [Crossref]
  9. J. Alda, “Paraxial optics,” in Encyclopedia of Optical Engineering, R. Driggers, ed. (Marcel Dekker, 2003), pp. 1920–1931.
  10. J. Alda and J. Arasa, “Paraxial ray tracing,” in Encyclopedia of Optical Engineering (Marcel Dekker, 2004).
  11. P. D. Lin and C. K. Sung, “Matrix-based paraxial skew ray-tracing in 3D systems with non-coplanar optical axis,” Optik 117, 329–340 (2006).
    [Crossref]
  12. N. Hagen and T. S. Tkaczyk, “Compound prism design principles I,” Appl. Opt. 50, 4998–5011 (2011).
    [Crossref]
  13. C. Y. Tsai and P. D. Lin, “Prism design based on image orientation change,” Appl. Opt. 45, 3951–3959 (2006).
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    [Crossref]
  15. P. D. Lin, “Sufficient conditions for the avoidance of spectral dispersion in optical prisms,” J. Opt. Soc. Am. A 33, 1257–1266 (2016).
    [Crossref]
  16. W. J. Smith, Modern Optical Engineering, 3rd ed. (Edmund Industrial Optics, 2001).
  17. F. J. Duarte and J. Piper, “Dispersion theory of multiple-prism beam expanders for pulsed dye lasers,” Opt. Commun. 43, 303–307 (1982).
    [Crossref]
  18. F. J. Duarte, “Ray transfer matrix analysis of multiple-prism dye laser oscillators,” Opt. Quantum Electron. 21, 47–54 (1989).
    [Crossref]
  19. F. J. Duarte, Tunable Laser Optics, 2nd ed. (CRC Press, 2015).
  20. F. J. Duarte, “Multiple-prism dispersion and 4 × 4 ray transfer matrices,” Opt. Quantum Electron. 24, 49–53 (1992).
    [Crossref]
  21. G. F. Marshall, “Risley prism scan patterns,” Proc. SPIE 3787, 74–86 (1999).
    [Crossref]
  22. Y. Li, “Third-order theory of the Risley-prism-based beam steering system,” Appl. Opt. 50, 679–686 (2011).
    [Crossref]
  23. A. Schitea, M. Tuef, and V. F. Duma, “Modeling of Risley prisms devices for exact scan patterns,” Proc. SPIE 8789, 878912 (2013).
    [Crossref]
  24. P. D. Lin, New Computation Methods for Geometrical Optics (Springer, 2013).
  25. R. L. Fork, O. E. Martinez, and J. P. Gordon, “Negative dispersion using pairs of prisms,” Opt. Lett. 9, 150–152 (1984).
    [Crossref]

2016 (1)

2013 (1)

A. Schitea, M. Tuef, and V. F. Duma, “Modeling of Risley prisms devices for exact scan patterns,” Proc. SPIE 8789, 878912 (2013).
[Crossref]

2011 (3)

2007 (1)

2006 (2)

C. Y. Tsai and P. D. Lin, “Prism design based on image orientation change,” Appl. Opt. 45, 3951–3959 (2006).
[Crossref]

P. D. Lin and C. K. Sung, “Matrix-based paraxial skew ray-tracing in 3D systems with non-coplanar optical axis,” Optik 117, 329–340 (2006).
[Crossref]

2001 (1)

J. P. Rolland, “The art of back-of-the-envelope paraxial raytracing,” IEEE Trans. Educ. 44, 365–372 (2001).
[Crossref]

1999 (1)

G. F. Marshall, “Risley prism scan patterns,” Proc. SPIE 3787, 74–86 (1999).
[Crossref]

1992 (1)

F. J. Duarte, “Multiple-prism dispersion and 4 × 4 ray transfer matrices,” Opt. Quantum Electron. 24, 49–53 (1992).
[Crossref]

1989 (1)

F. J. Duarte, “Ray transfer matrix analysis of multiple-prism dye laser oscillators,” Opt. Quantum Electron. 21, 47–54 (1989).
[Crossref]

1984 (1)

1982 (1)

F. J. Duarte and J. Piper, “Dispersion theory of multiple-prism beam expanders for pulsed dye lasers,” Opt. Commun. 43, 303–307 (1982).
[Crossref]

Alda, J.

J. Alda, “Paraxial optics,” in Encyclopedia of Optical Engineering, R. Driggers, ed. (Marcel Dekker, 2003), pp. 1920–1931.

J. Alda and J. Arasa, “Paraxial ray tracing,” in Encyclopedia of Optical Engineering (Marcel Dekker, 2004).

Arasa, J.

J. Alda and J. Arasa, “Paraxial ray tracing,” in Encyclopedia of Optical Engineering (Marcel Dekker, 2004).

Burch, J. M.

A. Gerrard and J. M. Burch, Introduction to Matrix Methods in Optics (Wiley, 1975).

Duarte, F. J.

F. J. Duarte, “Multiple-prism dispersion and 4 × 4 ray transfer matrices,” Opt. Quantum Electron. 24, 49–53 (1992).
[Crossref]

F. J. Duarte, “Ray transfer matrix analysis of multiple-prism dye laser oscillators,” Opt. Quantum Electron. 21, 47–54 (1989).
[Crossref]

F. J. Duarte and J. Piper, “Dispersion theory of multiple-prism beam expanders for pulsed dye lasers,” Opt. Commun. 43, 303–307 (1982).
[Crossref]

F. J. Duarte, Tunable Laser Optics, 2nd ed. (CRC Press, 2015).

Duma, V. F.

A. Schitea, M. Tuef, and V. F. Duma, “Modeling of Risley prisms devices for exact scan patterns,” Proc. SPIE 8789, 878912 (2013).
[Crossref]

Fork, R. L.

Freeman, M. H.

M. H. Freeman, Optics, 10th ed. (Butterworths, 1990).

Gerrard, A.

A. Gerrard and J. M. Burch, Introduction to Matrix Methods in Optics (Wiley, 1975).

Goodman, D. S.

D. S. Goodman, “General principles of geometrical optics,” in Handbook of Optics, M. Bass, ed. (McGraw-Hill, 1995), Vol. 1, Chap. 1.

Gordon, J. P.

Guenther, R.

R. Guenther, “Geometrical optics,” in Modern Optics (Wiley, 1990), pp. 129–212.

Hagen, N.

Hecht, E.

E. Hecht, Optics, 3rd ed. (Addison-Wesley, 1998).

Li, Y.

Lin, P. D.

Marshall, G. F.

G. F. Marshall, “Risley prism scan patterns,” Proc. SPIE 3787, 74–86 (1999).
[Crossref]

Martinez, O. E.

Pedrotti, F. L.

F. L. Pedrotti and S. L. Pedrotti, Introduction to Optics, 2nd ed. (Prentice-Hall, 1996).

Pedrotti, S. L.

F. L. Pedrotti and S. L. Pedrotti, Introduction to Optics, 2nd ed. (Prentice-Hall, 1996).

Piper, J.

F. J. Duarte and J. Piper, “Dispersion theory of multiple-prism beam expanders for pulsed dye lasers,” Opt. Commun. 43, 303–307 (1982).
[Crossref]

Rolland, J. P.

J. P. Rolland, “The art of back-of-the-envelope paraxial raytracing,” IEEE Trans. Educ. 44, 365–372 (2001).
[Crossref]

Schitea, A.

A. Schitea, M. Tuef, and V. F. Duma, “Modeling of Risley prisms devices for exact scan patterns,” Proc. SPIE 8789, 878912 (2013).
[Crossref]

Smith, W. J.

W. J. Smith, Modern Optical Engineering, 3rd ed. (Edmund Industrial Optics, 2001).

Sung, C. K.

P. D. Lin and C. K. Sung, “Matrix-based paraxial skew ray-tracing in 3D systems with non-coplanar optical axis,” Optik 117, 329–340 (2006).
[Crossref]

Tkaczyk, T. S.

Tsai, C. Y.

Tuef, M.

A. Schitea, M. Tuef, and V. F. Duma, “Modeling of Risley prisms devices for exact scan patterns,” Proc. SPIE 8789, 878912 (2013).
[Crossref]

Won, D. J.

D. J. Won, “Propagation of a general Gaussian beam by paraxial ray tracing in an arbitrary optical system,” J. Korean Phys. Soc. 58, 735–741 (2011).
[Crossref]

Appl. Opt. (4)

IEEE Trans. Educ. (1)

J. P. Rolland, “The art of back-of-the-envelope paraxial raytracing,” IEEE Trans. Educ. 44, 365–372 (2001).
[Crossref]

J. Korean Phys. Soc. (1)

D. J. Won, “Propagation of a general Gaussian beam by paraxial ray tracing in an arbitrary optical system,” J. Korean Phys. Soc. 58, 735–741 (2011).
[Crossref]

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

Opt. Commun. (1)

F. J. Duarte and J. Piper, “Dispersion theory of multiple-prism beam expanders for pulsed dye lasers,” Opt. Commun. 43, 303–307 (1982).
[Crossref]

Opt. Lett. (1)

Opt. Quantum Electron. (2)

F. J. Duarte, “Ray transfer matrix analysis of multiple-prism dye laser oscillators,” Opt. Quantum Electron. 21, 47–54 (1989).
[Crossref]

F. J. Duarte, “Multiple-prism dispersion and 4 × 4 ray transfer matrices,” Opt. Quantum Electron. 24, 49–53 (1992).
[Crossref]

Optik (1)

P. D. Lin and C. K. Sung, “Matrix-based paraxial skew ray-tracing in 3D systems with non-coplanar optical axis,” Optik 117, 329–340 (2006).
[Crossref]

Proc. SPIE (2)

G. F. Marshall, “Risley prism scan patterns,” Proc. SPIE 3787, 74–86 (1999).
[Crossref]

A. Schitea, M. Tuef, and V. F. Duma, “Modeling of Risley prisms devices for exact scan patterns,” Proc. SPIE 8789, 878912 (2013).
[Crossref]

Other (11)

P. D. Lin, New Computation Methods for Geometrical Optics (Springer, 2013).

F. J. Duarte, Tunable Laser Optics, 2nd ed. (CRC Press, 2015).

J. Alda, “Paraxial optics,” in Encyclopedia of Optical Engineering, R. Driggers, ed. (Marcel Dekker, 2003), pp. 1920–1931.

J. Alda and J. Arasa, “Paraxial ray tracing,” in Encyclopedia of Optical Engineering (Marcel Dekker, 2004).

W. J. Smith, Modern Optical Engineering, 3rd ed. (Edmund Industrial Optics, 2001).

A. Gerrard and J. M. Burch, Introduction to Matrix Methods in Optics (Wiley, 1975).

M. H. Freeman, Optics, 10th ed. (Butterworths, 1990).

R. Guenther, “Geometrical optics,” in Modern Optics (Wiley, 1990), pp. 129–212.

D. S. Goodman, “General principles of geometrical optics,” in Handbook of Optics, M. Bass, ed. (McGraw-Hill, 1995), Vol. 1, Chap. 1.

F. L. Pedrotti and S. L. Pedrotti, Introduction to Optics, 2nd ed. (Prentice-Hall, 1996).

E. Hecht, Optics, 3rd ed. (Addison-Wesley, 1998).

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Figures (10)

Fig. 1.
Fig. 1.

Schematic interpretation of homogeneous coordinate transformation matrix A ¯ g h .

Fig. 2.
Fig. 2.

Schematic representation of unit directional vector ¯ 0 originating from point source P ¯ 0 .

Fig. 3.
Fig. 3.

Schematic illustration showing paraxial ray propagating along straight-line path and then refracting at boundary surface r ¯ i .

Fig. 4.
Fig. 4.

Non-axially symmetrical system with multiple triangular prisms labeled sequentially from j = 1 to j = k .

Fig. 5.
Fig. 5.

Simple prism with apex angle η e 1 and refractive index ξ e 1 .

Fig. 6.
Fig. 6.

Percentage error distribution σ versus apex angle η e 1 .

Fig. 7.
Fig. 7.

Multiple-triangular-prism system used to obtain beam expansion factor 8 z / β 0 = 1 .

Fig. 8.
Fig. 8.

2D compound prism comprising k elements and n = 2 k flat boundary surfaces.

Fig. 9.
Fig. 9.

Rectangular prism.

Fig. 10.
Fig. 10.

Risley system composed of two rotational wedge prisms—in one of their four possible configurations [22].

Equations (49)

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A ¯ g h = [ I x J x K x t x I y J y K y t y I z J z K z t z 0 0 0 1 ] .
Δ R ¯ i g = [ Δ P ¯ i g Δ ¯ i g ] T = [ Δ P i x g Δ P i y g Δ P i z g Δ i x g Δ i x g Δ i x g ] T .
Δ R ¯ i h = [ Δ P ¯ i h Δ ¯ i h ] = [ I x J x K x 0 0 0 I y J y K y 0 0 0 I z J z K z 0 0 0 0 0 0 I x J x K x 0 0 0 I y J y K y 0 0 0 I z J z K z ] [ Δ P ¯ i g Δ ¯ i g ] = B ¯ g h [ Δ P ¯ i g Δ ¯ i g ] = B ¯ g h Δ R ¯ i g .
P ¯ 0 = [ P 0 x P 0 y P 0 z 1 ] T ,
¯ 0 = [ 0 x 0 y 0 z 0 ] = [ C β 0 C ( 90 ° + α 0 ) C β 0 S ( 90 ° + α 0 ) S β 0 0 ] ,
Δ R ¯ 0 = [ Δ P ¯ 0 Δ ¯ 0 ] = [ Δ P 0 x Δ P 0 y Δ P 0 z Δ 0 x Δ 0 y Δ 0 z ] T = [ 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 C β 0 S ( 90 ° + α 0 ) S β 0 C ( 90 ° + α 0 ) 0 0 0 C β 0 C ( 90 ° + α 0 ) S β 0 S ( 90 ° + α 0 ) 0 0 0 0 C β 0 ] [ Δ P 0 x Δ P 0 y Δ P 0 z Δ α 0 Δ β 0 ] .
[ Δ P 0 y Δ P 0 z Δ 0 y Δ 0 z ] = [ 1 0 0 0 1 0 0 0 S β 0 0 0 C β 0 ] [ Δ P 0 y Δ P 0 z Δ β 0 ] .
[ Δ P ¯ i i Δ ¯ i i ] = M ¯ i T ¯ i [ Δ P ¯ i 1 i Δ ¯ i 1 i ] ,
T ¯ i = [ 1 0 0 λ i , axis 0 0 0 1 0 0 λ i , axis 0 0 0 1 0 0 λ i , axis 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 ] .
M ¯ i = [ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 N i 0 0 0 0 0 0 N i 2 0 0 0 0 0 0 N i ] .
[ Δ P ¯ i i + 1 Δ ¯ i i + 1 ] = B ¯ i i + 1 [ Δ P ¯ i i Δ ¯ i i ] = B ¯ i i + 1 M ¯ i T ¯ i [ Δ P ¯ i 1 i Δ ¯ i 1 i ] .
B ¯ i i + 1 = [ a x b x c x 0 0 0 a y b y c y 0 0 0 a z b z c z 0 0 0 0 0 0 a x b x c x 0 0 0 a y b y c y 0 0 0 a z b z c z ]
A ¯ i i + 1 = A ¯ 0 i + 1 A ¯ i 0 = ( A ¯ i + 1 0 ) 1 ( A ¯ i 0 ) = [ a x b x c x d x a y b y c y d y a z b z c z d z 0 0 0 1 ] .
[ Δ P ¯ 2 j 1 2 j Δ ¯ 2 j 1 2 i ] = B ¯ 2 j 1 2 j M ¯ 2 j 1 T ¯ 2 j 1 [ Δ P ¯ 2 j 2 2 j 1 Δ ¯ 2 j 2 2 j 1 ] = B ¯ 2 j 1 2 j M ¯ 2 j 1 T ¯ 2 j 1 ( B ¯ 2 j 2 2 j 1 [ Δ P ¯ 2 j 2 2 j 2 Δ ¯ 2 j 2 2 j 2 ] ) .
[ Δ P ¯ 2 j 2 j Δ ¯ 2 j 2 j ] = M ¯ 2 j T ¯ 2 j [ Δ P ¯ 2 j 1 2 j Δ ¯ 2 j 1 2 j ] ,
[ Δ P ¯ 2 j 2 j Δ ¯ 2 j 2 j ] = ( M ¯ 2 j T ¯ 2 j B ¯ 2 j 1 2 j M ¯ 2 j 1 ) ( T ¯ 2 j 1 B ¯ 2 j 2 2 j 1 ) [ Δ P ¯ 2 j 2 2 j 2 Δ ¯ 2 j 2 2 j 2 ] M ¯ e j T ¯ e j [ Δ P ¯ 2 j 2 2 j 2 Δ ¯ 2 j 2 2 j 2 ] ,
[ Δ P 2 j y 2 j Δ P 2 j z 2 j Δ 2 j y 2 j Δ 2 j z 2 j ] = M ¯ e j T ¯ e j [ Δ P 2 j 2 y 2 j 2 Δ P 2 j 2 z 2 j 2 Δ 2 j 2 y 2 j 2 Δ 2 j 2 z 2 j 2 ] .
T ¯ e j = [ C φ j S φ j C φ j λ 2 j 1 , axis S φ j λ 2 j 1 , axis S φ j C φ j S φ j λ 2 j 1 , axis C φ j λ 2 j 1 , axis 0 0 C φ j S φ j 0 0 S φ j C φ j ] ,
M ¯ e j = [ 0 0 0 0 0 C η e j S η e j λ 2 j , axis N 2 j 1 2 C η e j λ 2 j , axis N 2 j 1 0 0 C η e j N 2 j 1 2 N 2 j 2 S η e j N 2 j 1 N 2 j 2 0 0 S η e j N 2 j 1 2 N 2 j C η e j N 2 j 1 N 2 j ] [ 0 0 ¯ 1 × 3 0 ¯ 3 × 1 M ¯ e j ] .
T ¯ e j = [ C φ j S φ j C φ j v j S φ j v j S φ j C φ j S φ j v j C φ j v j 0 0 C φ j S φ j 0 0 S φ j C φ j ] ,
M ¯ e j = [ 0 0 0 0 0 C η e j S η e j N e j 2 q e j C η e j N e j q e j 0 0 C η e j S η e j / N e j 0 0 S η e j N e j C η e j ] .
[ Δ P n y n Δ P n z n Δ n y n Δ n z n ] = [ Δ P 2 k y 2 k Δ P 2 k z 2 k Δ 2 k y 2 k Δ 2 k z 2 k ] = M ¯ e k T ¯ e k [ Δ P 2 k 2 y 2 k 2 Δ P 2 k 2 z 2 k 2 Δ 2 k 2 y 2 k 2 Δ 2 k 2 z 2 k 2 ] = M ¯ e k T ¯ e k M ¯ e k 1 T ¯ e k 1 [ Δ P 2 k 4 y 2 k 4 Δ P 2 k 4 z 2 k 4 Δ 2 k 4 y 2 k 4 Δ 2 k 4 z 2 k 4 ] = = M ¯ e k T ¯ e k M ¯ e j T ¯ e j M ¯ e 1 T ¯ e 1 [ Δ P 0 y Δ P 0 z Δ 0 y Δ 0 z ] .
M ¯ e j = [ 0 [ 0 0 0 ] [ 0 0 0 ] [ C η e j S η e j N e j 2 q e j C η e j N e j q e j 0 C η e j S η e j / N e j 0 S η e j N e j C η e j ] ] ,
T ¯ e j = [ C φ j [ S φ j C φ j v j S φ j v j ] [ S φ j 0 0 ] [ C φ j S φ j v j C φ j v j 0 C φ j S φ j 0 S φ j C φ j ] ] .
[ Δ P n z n Δ n y n Δ n z n ] = M ¯ e k T ¯ e k M ¯ e j T ¯ e j M ¯ e 1 T ¯ e 1 [ Δ P 0 z Δ 0 y Δ 0 z ] ( C η e k S φ k ) ( C η e j S φ j ) ( C η e 1 S φ 1 ) [ Δ P 0 y 0 0 ] ,
M ¯ e j = [ C η e j S η e j N e j 2 q e j C η e j N e j q e j 0 C η e j S η e j / N e j 0 S η e j N e j C η e j ] ( j = 1    to    j = k ) ,
T ¯ e j = [ C φ j S φ j v j C φ j v j 0 C φ j S φ j 0 S φ j C φ j ] ( j = 1    to    j = k ) .
[ Δ P n z n Δ n y n Δ n z n ] = M ¯ e k T ¯ e k M ¯ e j T ¯ e j M ¯ e 1 T ¯ e 1 [ 1 0 0 S β 0 0 C β 0 ] [ Δ P 0 z Δ β 0 ] ( C η e k S φ k ) ( C η e j S φ j ) ( C η e 1 S φ 1 ) [ Δ P 0 y 0 0 ] .
[ Δ 2 y 2 Δ 2 z 2 ] = [ C η e 1 S η e 1 / N e 1 S η e 1 N e 1 C η e 1 ] [ S β 0 C β 0 ] Δ β 0 ,
Δ 2 z = [ S η e 1 C η e 1 ] [ Δ 2 y 2 Δ 2 z 2 ] = [ S η e 1 C η e 1 ] [ S β 0 C η e 1 + C β 0 S η e 1 / N e 1 S β 0 S η e 1 N e 1 + C β 0 C η e 1 ] Δ β 0 .
[ P ¯ 0 ¯ 0 ] = [ 0 50 10 0 C β 0 S β 0 ] T .
M ¯ e 1 = [ C η e 1 0 0 0 C η e 1 S η e 1 / N e 1 0 S η e 1 N e 1 C η e 1 ] ,
M ¯ e 2 = [ C η e 1 0 0 0 C η e 1 S η e 1 / N e 1 0 S η e 1 N e 1 C η e 1 ] .
T ¯ e 1 = [ C φ 1 S φ 1 v 1 C φ 1 v 1 0 C φ 1 S φ 1 0 S φ 1 C φ 1 ] ,
T ¯ e 2 = T ¯ e 4 = [ 1 0 v 2 0 1 0 0 0 1 ] ,
T ¯ e 3 = [ C 2 φ 1 S 2 φ 1 v 3 C 2 φ 1 v 3 0 C 2 φ 1 S 2 φ 1 0 S 2 φ 1 C 2 φ 1 ] .
[ Δ P 8 z 8 Δ 8 y 8 Δ 8 z 8 ] = M ¯ e 4 T ¯ e 4 M ¯ e 3 T ¯ e 3 M ¯ e 2 T ¯ e 2 M ¯ e 1 T ¯ e 1 [ 1 0 0 0 0 1 ] [ Δ P 0 z Δ β 0 ] = [ C η e 1 4 C 2 φ 1 C φ 1 Δ P 0 z + C η e 1 2 C φ 1 ( C η e 1 2 C 2 φ 1 v 1 + v 3 ) Δ β 0 S φ 1 Δ β 0 C φ 1 Δ β 0 ] .
[ Δ P 8 y Δ P 8 z ] = [ C φ 1 S φ 1 S φ 1 C φ 1 ] [ Δ P 8 y 8 Δ P 8 z 8 ] ,
[ Δ 8 y Δ 8 z ] = [ C φ 1 S φ 1 S φ 1 C φ 1 ] [ Δ 8 y 8 Δ 8 z 8 ] = [ C φ 1 S φ 1 S φ 1 C φ 1 ] [ S φ 1 Δ β 0 C φ 1 Δ β 0 ] = [ 0 1 ] Δ β 0 .
[ Δ P 2 z 2 Δ 2 y 2 Δ 2 z 2 ] = M ¯ e 1 T ¯ e 1 [ 1 0 0 S β 0 0 C β 0 ] [ Δ P 0 z Δ β 0 ] C η e 1 S φ 1 [ Δ P 0 y 0 0 ] ( k = 1 ) ,
[ Δ P n z n Δ n y n Δ n z n ] = M ¯ e k M ¯ e k 1 M ¯ e j M ¯ e 2 M ¯ e 1 T ¯ e 1 [ 1 0 0 S β 0 0 C β 0 ] [ Δ P 0 z Δ β 0 ] ( k 2 ) .
M ¯ e j = [ C η e j S η e j q e j N e j 2 C η e j q e j N e j 0 C η e j N e j 2 N e j + 1 2 S η e j N e j N e j + 1 2 0 S η e j N e j 2 N e j + 1 C η e j N e j N e j + 1 ] .
M ¯ e j = [ 1 0 q e j N e j 0 N e j 2 N e j + 1 2 0 0 0 N e j N e j + 1 ] .
M ¯ e j = [ 1 0 q e j N e j 0 1 0 0 0 1 ] .
[ Δ P n z n Δ n y n Δ n z n ] = M ¯ e k T ¯ e k M ¯ e j T ¯ e j M ¯ e 1 T ¯ e 1 [ 1 0 0 S β 0 0 C β 0 ] [ Δ P 0 z Δ β 0 ] S φ k S φ j S φ 1 [ Δ P 0 y 0 0 ] .
[ Δ P 2 z 2 Δ 2 y 2 Δ 2 z 2 ] = M ¯ e 1 T ¯ e 1 [ Δ P 0 z S β 0 Δ β 0 C β 0 Δ β 0 ] S φ 1 [ Δ P 0 y 0 0 ] = [ C ( φ 1 β 0 ) ( v 1 + N e 1 q e 1 ) Δ β 0 S φ 1 Δ P 0 y + C φ 1 Δ P 0 z S ( φ 1 β 0 ) Δ β 0 C ( φ 1 β 0 ) Δ β 0 ] .
[ Δ 2 y Δ 2 z ] = [ C φ 1 S φ 1 S φ 1 C φ 1 ] [ Δ 2 y 2 Δ 2 z 2 ] = [ C φ 1 S φ 1 S φ 1 C φ 1 ] [ S φ 1 C φ 1 ] Δ β 0 = [ 0 1 ] Δ β 0 .
[ Δ P 5 z 5 Δ 5 y 5 Δ 5 z 5 ] = T ¯ e 3 M ¯ e 2 T ¯ e 2 M ¯ e 1 T ¯ e 1 [ 1 0 0 0 0 1 ] [ Δ P 0 z Δ β 0 ] .
Δ P 5 z 5 = C η e 1 C η e 2 C ( η e 1 + η e 2 ) Δ P 0 z + ( N e 1 q e 1 + v 1 ) C η e 1 C η e 2 C ( η e 1 + η e 2 ) Δ β 0 + [ ( N e 2 q e 2 + v 2 + v 3 ) C η e 2 S ( η e 1 + η e 2 ) ( N e 2 q e 2 + v 3 ) N e 2 S η e 2 C ( η e 1 + η e 2 ) ] S η e 1 Δ β 0 / N e 1 + [ ( N e 2 q e 2 + v 2 + v 3 ) C η e 2 C ( η e 1 + η e 2 ) + ( N e 2 q e 2 + v 3 ) N e 2 S η e 2 S ( η e 1 + η e 2 ) ] C η e 1 Δ β 0 .

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