Abstract

The design of optical systems containing prisms is comparatively difficult since each prism may contain multiple boundary surfaces. Many geometrical optical merit functions have been proposed based on first-order derivatives of the geometrical quantities of the system with respect to the boundary variable vector Xi. However, transferring the computed quantities into the system variable vector Xsys is still highly challenging. Accordingly, this study proposes a new numerical method for determining the Jacobian matrix between Xi and Xsys directly. The proposed methodology can be easily implemented in computer code and provides a potential basis for the future development of a numerical technique for computing the second-order derivatives of the geometrical quantities of an optical system.

© 2011 Optical Society of America

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References

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  1. Optical Research Associates, http://www.opticalres.com.
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  14. R. P. Paul, Robot Manipulators—Mathematics, Programming and Control (MIT, 1982).

2008

P. D. Lin and C. Y. Tsai, “General method for determining the first order gradients of skew rays of optical systems with non-coplanar optical axes,” Appl. Phys. B , 91, 621–628(2008).
[CrossRef]

2007

1999

R. Shi and J. Kross, “Differential ray tracing for optical design,” Proc. SPIE 3737, 149–160 (1999).
[CrossRef]

1997

1989

J. Kross, “Differential ray tracing formulae for optical calculations: principles and applications,” Proc. SPIE 1013, 10–18(1989).

1988

W. Oertmann, “Differential ray tracing formulae; applications especially to aspheric optical systems,” Proc. SPIE 1013, 20–26(1988).

1985

1982

1980

W. Mandler, “Uber die Berechnung einfacher GauB-Objektive,” II, Beispiel 5, Optik 55, 219–240 (1980).

1976

1968

1957

Andersen, T. B.

Feder, D. P.

Forbes, G. W.

Kross, J.

R. Shi and J. Kross, “Differential ray tracing for optical design,” Proc. SPIE 3737, 149–160 (1999).
[CrossRef]

J. Kross, “Differential ray tracing formulae for optical calculations: principles and applications,” Proc. SPIE 1013, 10–18(1989).

Lin, P. D.

P. D. Lin and C. Y. Tsai, “General method for determining the first order gradients of skew rays of optical systems with non-coplanar optical axes,” Appl. Phys. B , 91, 621–628(2008).
[CrossRef]

P. D. Lin and C. Y. Tsai, “First order gradients of skew rays of axis-symmetrical optical systems,” J. Opt. Soc. Am. A 24, 776–784 (2007).
[CrossRef]

Mandler, W.

W. Mandler, “Uber die Berechnung einfacher GauB-Objektive,” II, Beispiel 5, Optik 55, 219–240 (1980).

Oertmann, W.

W. Oertmann, “Differential ray tracing formulae; applications especially to aspheric optical systems,” Proc. SPIE 1013, 20–26(1988).

Paul, R. P.

R. P. Paul, Robot Manipulators—Mathematics, Programming and Control (MIT, 1982).

Shi, R.

R. Shi and J. Kross, “Differential ray tracing for optical design,” Proc. SPIE 3737, 149–160 (1999).
[CrossRef]

Stavroudis, O.

Stone, B. D.

Tsai, C. Y.

P. D. Lin and C. Y. Tsai, “General method for determining the first order gradients of skew rays of optical systems with non-coplanar optical axes,” Appl. Phys. B , 91, 621–628(2008).
[CrossRef]

P. D. Lin and C. Y. Tsai, “First order gradients of skew rays of axis-symmetrical optical systems,” J. Opt. Soc. Am. A 24, 776–784 (2007).
[CrossRef]

Appl. Opt.

Appl. Phys. B

P. D. Lin and C. Y. Tsai, “General method for determining the first order gradients of skew rays of optical systems with non-coplanar optical axes,” Appl. Phys. B , 91, 621–628(2008).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Optik

W. Mandler, “Uber die Berechnung einfacher GauB-Objektive,” II, Beispiel 5, Optik 55, 219–240 (1980).

Proc. SPIE

R. Shi and J. Kross, “Differential ray tracing for optical design,” Proc. SPIE 3737, 149–160 (1999).
[CrossRef]

J. Kross, “Differential ray tracing formulae for optical calculations: principles and applications,” Proc. SPIE 1013, 10–18(1989).

W. Oertmann, “Differential ray tracing formulae; applications especially to aspheric optical systems,” Proc. SPIE 1013, 20–26(1988).

Other

Optical Research Associates, http://www.opticalres.com.

R. P. Paul, Robot Manipulators—Mathematics, Programming and Control (MIT, 1982).

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Figures (14)

Fig. 1
Fig. 1

Illustrative optical system containing one double-convex lens, two prisms, and an image plane.

Fig. 2
Fig. 2

Skew ray tracing at a flat boundary surface.

Fig. 3
Fig. 3

Skew ray tracing at a spherical boundary surface.

Fig. 4
Fig. 4

A g h = tran ( t x , 0 , 0 ) , translation about the x h axis through distance t x .

Fig. 5
Fig. 5

A g h = tran ( 0 , t y , 0 ) , translation about the y h axis through distance t y .

Fig. 6
Fig. 6

A g h = tran ( 0 , 0 , t z ) , translation about the z h axis through distance t z .

Fig. 7
Fig. 7

A g h = xrot ( ω x ) , rotation about the x h axis through angle ω x .

Fig. 8
Fig. 8

A g h = yrot ( ω y ) , rotation about the y h axis through angle ω y .

Fig. 9
Fig. 9

A g h = zrot ( ω z ) , rotation about the z h axis through angle ω z .

Fig. 10
Fig. 10

Element j = 1 , double-convex lens.

Fig. 11
Fig. 11

Element j = 2 is a right-angle prism if σ e 2 = 90 ° .

Fig. 12
Fig. 12

Element j = 3 , prism with roof.

Fig. 13
Fig. 13

Element j = 4 , image plane.

Fig. 14
Fig. 14

Representation of unit directional vector 0 0 originating from source point P 0 0 .

Tables (2)

Tables Icon

Table 1 Values of the Six Pose Variables for Boundary Surfaces in the System of Fig. 1

Tables Icon

Table 2 Values of ( OPL e 2 ) / α 0 from the FD Method for Different Δ α 0 a

Equations (100)

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A 0 i = tran ( t i x , 0 , 0 ) tran ( 0 , t i y , 0 ) tran ( 0 , 0 , t i z ) zrot ( ω i z ) yrot ( ω i y ) xrot ( ω i x ) = [ I i x J i x K i x t i x I i y J i y K i y t i y I i z J i z K i z t i z 0 0 0 1 ] = [ C ω i z C ω i y C ω i z S ω i y S ω i x S ω i z C ω i x C ω i z S ω i y C ω i x + S ω i z S ω i x t i x S ω i z C ω i y S ω i z S ω i y S ω i x + C ω i z C ω i x S ω i z S ω i y C ω i x C ω i z S ω i x t i y S ω i y C ω i y S ω i x C ω i y C ω i x t i z 0 0 0 1 ] ,
tran ( ε 1 , 0 , 0 ) = [ 1 0 0 ε 1 0 1 0 0 0 0 1 0 0 0 0 1 ] ,
tran ( 0 , ε 2 , 0 ) = [ 1 0 0 0 0 1 0 ε 2 0 0 1 0 0 0 0 1 ] ,
tran ( 0 , 0 , ε 3 ) = [ 1 0 0 0 0 1 0 0 0 0 1 ε 3 0 0 0 1 ] ,
xrot ( ε 4 ) = [ 1 0 0 0 0 C ε 4 S ε 4 0 0 S ε 4 C ε 4 0 0 0 0 1 ] ,
yrot ( ε 5 ) = [ C ε 5 0 S ε 5 0 0 1 0 0 S ε 5 0 C ε 5 0 0 0 0 1 ] ,
zrot ( ε 6 ) = [ C ε 6 S ε 6 0 0 S ε 6 C ε 6 0 0 0 0 1 0 0 0 0 1 ] .
X i = [ t i x t i y t i z ω i x ω i y ω i z ξ i 1 ξ i R i ] T
X e 0 = [ P 0 x 0 P 0 y 0 P 0 z 0 β 0 α 0 ] T .
A ej 0 = tran ( t ej x 0 , t ej y 0 , t ej z 0 ) zrot ( ω ej z 0 ) yrot ( ω ej y 0 ) xrot ( ω ej x 0 ) = [ I ej x 0 J ej x 0 K ej x 0 t ej x 0 I ej y 0 J ej y 0 K ej y 0 t ej y 0 I ej z 0 J ej z 0 K ej z 0 t ej z 0 0 0 0 1 ] .
X e 1 = [ t e 1 x 0 t e 1 y 0 t e 1 z 0 ω e 1 x 0 ω e 1 y 0 ω e 1 z 0 ξ e 1 ξ air R 1 q e 1 R 2 ] T .
X e 2 = [ t e 2 x 0 t e 2 y 0 t e 2 z 0 ω e 2 x 0 ω e 2 y 0 ω e 2 z 0 ξ e 2 ξ glue ξ air h e 2 w e 2 g e 2 m e 2 ψ e 2 σ e 2 ] T .
X e 3 = [ t e 3 x 0 t e 3 y 0 t e 3 z 0 ω e 3 x 0 ω e 3 y 0 ω e 3 z 0 ξ air ξ e 3 ξ glue h e 3 w e 3 g e 3 ψ e 3 σ e 3 ] T .
X e 4 = [ t e 4 x 0 t e 4 y 0 t e 4 z 0 ω e 4 x 0 ω e 4 y 0 ω e 4 z 0 ] T .
X e 1 = [ t e 1 x 0 v 1 + R 1 t e 1 z 0 ω e 1 x 0 ω e 1 y 0 ω e 1 z 0 ξ e 1 ξ air R 1 q e 1 R 2 ] T ,
X e 2 = [ w e 2 / 2 v 1 + q e 1 + v 2 g e 2 ω e 2 x 0 ω e 2 y 0 ω e 2 z 0 ξ e 2 ξ glue ξ air h e 2 w e 2 g e 2 m e 2 ψ e 2 σ e 2 ] T ,
X e 3 = [ w e 2 / 2 v 1 + q e 1 + v 2 + h e 2 + 2 gap glue g e 2 ψ e 2 σ e 3 ω e 3 y 0 ω e 3 z 0 ξ air ξ e 3 ξ glue h e 3 w e 2 g e 2 ψ e 3 σ e 3 ] T ,
X e 4 = [ t e 4 x 0 v 1 + q e 1 + v 2 + h e 2 + 2 gap glue + v 4 t e 4 z 0 ω e 4 x 0 ω e 4 y 0 ω e 4 z 0 ] T .
X sys = [ P 0 x 0 P 0 y 0 P 0 z 0 β 0 α 0 t e 1 x 0 v 1 t e 1 z 0 ω e 1 x 0 ω e 1 y 0 ω e 1 z 0 ξ e 1 ξ air R 1 q e 1 R 2 v 2 ω e 2 x 0 ω e 2 y 0 ω e 2 z 0 ξ e 2 ξ glue h e 2 w e 2 g e 2 m e 2 ψ e 2 σ e 2 ω e 3 y 0 ω e 3 z 0 ξ e 3 gap glue h e 3 ψ e 3 σ e 3 t e 4 x 0 v 4 t e 4 z 0 ω e 4 x 0 ω e 4 y 0 ω e 4 z 0 ] T .
A 0 i = ( A i 0 ) 1 = ( A ej 0 A i ej ) 1 = A ej i A 0 ej = ( A i ej ) 1 ( A ej 0 ) 1 ,
A 1 e 1 = I 4 × 4 ,
A 2 e 1 = tran ( 0 , R 1 + q e 1 R 2 , 0 ) ,
A 3 e 2 = tran ( 0 , 0 , m e 2 ) ,
A 4 e 2 = tran ( 0 , 0 , m e 2 ) xrot ( ψ e 2 ) ,
A 5 e 2 = xrot ( 180 ° σ e 2 ) ,
A 6 e 2 = A 4 e 2 ,
A 7 e 3 = xrot ( σ e 3 ) ,
A 8 e 3 = I 4 × 4 ,
A 9 e 3 = tran ( w e 2 , 0 , h e 3 ) yrot ( 45 ° ) zrot ( ψ e 3 ) ,
A 10 e 3 = tran ( 0 , 0 , h e 3 ) yrot ( 45 ° ) zrot ( ψ e 3 ) ,
A 11 e 3 = A 7 e 3 ,
A 12 e 3 = A 8 e 3 ,
A 13 e 4 = I 4 × 4 .
A 0 1 = A e 1 1 A 0 e 1 = xrot ( ω e 1 x 0 ) yrot ( ω e 1 y 0 ) zrot ( ω e 1 z 0 ) tran ( t e 1 x 0 , v 1 R 1 , t e 1 z 0 ) ,
A 0 2 = A e 1 2 A 0 e 1 = tran ( 0 , R 1 q e 1 + R 2 , 0 ) xrot ( ω e 1 x 0 ) yrot ( ω e 1 y 0 ) zrot ( ω e 1 z 0 ) tran ( t e 1 x 0 , v 1 R 1 , t e 1 z 0 ) ,
A 0 3 = A e 2 3 A 0 e 2 = tran ( 0 , 0 , m e 2 ) xrot ( ω e 2 x 0 ) yrot ( ω e 2 y 0 ) zrot ( ω e 2 z 0 ) tran ( 0.5 w , e 2 v 1 q e 1 v 2 , g e 2 ) ,
A 0 4 = A e 2 4 A 0 e 2 = xrot ( ψ e 2 ) tran ( 0 , 0 , m e 2 ) xrot ( ω e 2 x 0 ) yrot ( ω e 2 y 0 ) zrot ( ω e 2 z 0 ) tran ( 0.5 w , e 2 v 1 q e 1 v 2 , g e 2 ) ,
A 0 5 = A e 2 5 A 0 e 2 = xrot ( 180 ° + σ e 2 ) xrot ( ω e 2 x 0 ) yrot ( ω e 2 y 0 ) zrot ( ω e 2 z 0 ) tran ( 0.5 w , e 2 v 1 q e 1 v 2 , g e 2 ) ,
A 0 6 = A 0 4 ,
A 0 7 = A e 3 7 A 0 e 3 = xrot ( σ e 3 ) xrot ( ψ e 2 + σ e 3 ) yrot ( ω e 3 y 0 ) zrot ( ω e 3 z 0 ) tran ( 0.5 w , e 2 v 1 q e 1 v 2 h e 2 2 gap glue , g e 2 ) ,
A 0 8 = A e 3 8 A 0 e 3 = xrot ( ψ e 2 + σ e 3 ) yrot ( ω e 3 y 0 ) zrot ( ω e 3 z 0 ) tran ( 0.5 w , e 2 v 1 q e 1 v 2 h e 2 2 gap glue , g e 2 ) ,
A 0 9 = A e 3 9 A 0 e 3 = zrot ( ψ e 3 ) yrot ( 45 ° ) tran ( w e 2 , 0 , h e 3 ) xrot ( ψ e 2 + σ e 3 ) yrot ( ω e 3 y 0 ) zrot ( ω e 3 z 0 ) tran ( 0.5 w , e 2 v 1 q e 1 v 2 h e 2 2 gap glue , g e 2 ) ,
A 0 10 = A e 3 10 A 0 e 3 = zrot ( ψ e 3 ) yrot ( 45 ° ) tran ( 0 , 0 , h e 3 ) xrot ( ψ e 2 + σ e 3 ) yrot ( ω e 3 y 0 ) zrot ( ω e 3 z 0 ) tran ( 0.5 w , e 2 v 1 q e 1 v 2 h e 2 2 gap glue , g e 2 ) ,
A 0 11 = A 0 7 ,
A 0 12 = A 0 8 ,
A 0 13 = A e 4 13 A 0 e 4 = xrot ( ω e 4 x 0 ) yrot ( ω e 4 y 0 ) zrot ( ω e 4 z 0 ) tran ( t e 4 x 0 , v 1 q e 1 v 2 h e 2 2 gap glue v 4 , t e 4 z 0 ) .
ω i z = atan 2 ( I i y , I i x ) ,
ω i y = atan 2 ( I i z , I i x C ω i z + I i y S ω i z ) ,
ω i x = atan 2 ( K i x S ω i z K i y C ω i z , J i x S ω i z + J i y C ω i z ) ,
A 0 i = Π v = 1 v = p operator v ( ε v ) = operator 1 ( ε 1 ) operator 2 ( ε 2 ) ... operator v 1 ( ε v 1 ) operator v ( ε v ) operator v + 1 ( ε v + 1 ) ... operator p 1 ( ε p 1 ) operator p ( ε p ) ,
A 0 i ε v = [ I i x / ε v J i x / ε v K i x / ε v t i x / ε v I i y / ε v J i y / ε v K i y / ε v t i y / ε v I i z / ε v J i z / ε v K i z / ε v t i z / ε v 0 0 0 0 ] = operator 1 ( ε 1 ) operator 2 ( ε 2 ) ... operator v 1 ( ε v 1 ) Doperator v ( ε v ) operator v + 1 ( ε v + 1 ) ... operator p 1 ( ε p 1 ) operator p ( ε p ) ,
Doperator v = Dtran ( ε v , 0 , 0 ) = [ 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 ] = Dtran ( 1 , 0 , 0 ) .
Doperator v = Dtran ( 0 , ε v , 0 ) = [ 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 ] = Dtran ( 0 , 1 , 0 ) .
Doperator v = Dtran ( 0 , 0 , ε v ) = [ 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 ] = Dtran ( 0 , 0 , 1 ) .
Doperator v = Dxrot ( ε v ) = [ 0 0 0 0 0 S ε v C ε v 0 0 C ε v S ε v 0 0 0 0 0 ] .
Doperator v = Dyrot ( ε v ) = [ S ε v 0 C ε v 0 0 0 0 0 C ε v 0 S ε v 0 0 0 0 0 ] .
Doperator v = Dzrot ( ε v ) = [ S ε v C ε v 0 0 C ε v S ε v 0 0 0 0 0 0 0 0 0 0 ] .
v = 1 v = p A 0 i ε v d ε v = v = 1 v = p operator 1 ( ε 1 ) operator 2 ( ε 2 ) ... operator v 1 ( ε v 1 ) Doperator v ( ε v ) d ε v operator v + 1 ( ε v + 1 ) ... operator p 1 ( ε p 1 ) operator p ( ε p ) .
J i ( u , v ) = X i / X sys = [ ( t i x , t i y , t i z ) X e 0 ( t i x , t i y , t i z ) X ξ ( t i x , t i y , t i z ) X R ( t i x , t i y , t i z ) X rest ( ω i x , ω i y , ω i z ) X e 0 ( ω i x , ω i y , ω i z ) X ξ ( ω i x , ω i y , ω i z ) X R ( ω i x , ω i y , ω i z ) X rest ( ξ i 1 , ξ i ) X e 0 ( ξ i 1 , ξ i ) X ξ ( ξ i 1 , ξ i ) X R ( ξ i 1 , ξ i ) X rest R i X e 0 R i X ξ R i X R R i X rest ] 9 × 41 ,
( t i x , t i y , t i z ) X e 0 = 0 3 × 5 ,
( ω i x , ω i y , ω i z ) X e 0 = 0 3 × 5 ,
( ξ i 1 , ξ i ) X e 0 = 0 2 × 5 ,
R i X e 0 = 0 1 × 5 .
( ξ i 1 , ξ i ) / X rest = 0 2 × 30 .
( t i x , t i y , t i z ) / X ξ = ( ω i x , ω i y , ω i z ) / X ξ = 0 3 × 5 ,
( ω i x , ω i y , ω i z ) / X R = 0 3 × 2 .
R i / X ξ = 0 1 × 5 ,
( ξ i 1 , ξ i ) / X R = 0 2 × 2 .
R i / X rest = 0 1 × 30 .
J i ( 7 , v ) = ξ i 1 ξ = { 1 ξ i 1 = ξ 0 ξ i 1 ξ ,
J i ( 8 , v ) = ξ i ξ = { 1 ξ i = ξ 0 ξ i ξ .
J 1 ( 7 , 13 ) = J 2 ( 7 , 12 ) = J 3 ( 7 , 13 ) = J 4 ( 7 , 21 ) = J 5 ( 7 , 21 ) = J 6 ( 7 , 21 ) = J 7 ( 7 , 22 ) = J 8 ( 7 , 31 ) = J 9 ( 7 , 31 ) = J 10 ( 7 , 31 ) = J 11 ( 7 , 31 ) = J 12 ( 7 , 31 ) = J 13 ( 7 , 13 ) = 1 ,
J 1 ( 8 , 12 ) = J 2 ( 8 , 13 ) = J 3 ( 8 , 21 ) = J 4 ( 8 , 21 ) = J 5 ( 8 , 21 ) = J 6 ( 8 , 22 ) = J 7 ( 8 , 31 ) = J 8 ( 8 , 31 ) = J 9 ( 8 , 31 ) = J 10 ( 8 , 31 ) = J 11 ( 8 , 31 ) = J 12 ( 8 , 13 ) = J 13 ( 8 , 13 ) = 1 ,
J i ( 9 , v ) = R i R = { 1 R i = R 0 R i R .
A 0 i ε v = [ I i x / ε v J i x / ε v K i x / ε v t i x / ε v I i y / ε v J i y / ε v K i y / ε v t i y / ε v I i z / ε v J i z / ε v K i z / ε v t i z / ε v 0 0 0 0 ] ,
I i x / ε v = S ω i z C ω i y ω i z / ε v C ω i z S ω i y ω i y / ε v ,
I i y / ε v = C ω i z C ω i y ω i z / ε v S ω i z S ω i y ω i y / ε v ,
I i z / ε v = C ω i y ω i y / ε v ,
J i x / ε v = S ω i z S ω i y S ω i x ω i z / ε v + C ω i z C ω i y S ω i x ω i y / ε v + C ω i z S ω i y C ω i x ω i x / ε v C ω i z C ω i x ω i z / ε v + S ω i z S ω i x ω i x / ε v ,
J i y / ε v = S ω i z S ω i y S ω i x ω i z / ε v + C ω i z C ω i y S ω i x ω i y / ε v + C ω i z S ω i y C ω i x ω i x / ε v C ω i z C ω i x ω i z / ε v + S ω i z S ω i x ω i x / ε v ,
J i z / ε v = S ω i y S ω i x ω i y / ε v + C ω i y C ω i x ω i x / ε v ,
K i x / ε v = S ω i z S ω i y C ω i x ω i z / ε v + C ω i z C ω i y C ω i x ω i y / ε v C ω i z S ω i y S ω i x ω i x / ε v + C ω i z S ω i x ω i z / ε v + S ω i z C ω i x ω i x / ε v ,
K i y / ε v = C ω i z S ω i y C ω i x ω i z / ε v + S ω i z C ω i y C ω i x ω i y / ε v S ω i z S ω i y S ω i x ω i x / ε v + S ω i z S ω i x ω i z / ε v C ω i z C ω i x ω i x / ε v ,
K i y / ε v = S ω i y C ω i x ω i y / ε v C ω i y S ω i x ω i x / ε v .
( t i x , t i y , t i z ) / X R = v = 1 41 ( t i x , t i y , t i z ) / ε v ,
( t i x , t i y , t i z ) / X rest = v = 1 41 ( t i x , t i y , t i z ) / ε v ,
( ω i x , ω i y , ω i z ) / X rest = v = 1 41 ( ω i x , ω i y , ω i z ) / ε v .
J 9 ( 1 , 7 ) = 1 , J 9 ( 1 , 15 ) = 1 , J 9 ( 1 , 17 ) = 1 , J 9 ( 1 , 23 ) = 1 , J 9 ( 1 , 27 ) = 5 , J 9 ( 1 , 30 ) = 5 , J 9 ( 1 , 32 ) = 1.4141 ,
J 9 ( 1 , 34 ) = 10 , J 9 ( 1 , 35 ) = 5 , J 9 ( 2 , 24 ) = 0.3536 , J 9 ( 2 , 25 ) = 0.7071 , J 9 ( 2 , 27 ) = 40.1909 ,
J 9 ( 2 , 30 ) = 40.1909 , J 9 ( 2 , 33 ) = 0.7071 , J 9 ( 2 , 34 ) = 56.8385 , J 9 ( 2 , 35 ) = 40.1909 , J 9 ( 3 , 24 ) = 0.3536 ,
J 9 ( 3 , 25 ) = 0.7071 , J 9 ( 3 , 27 ) = 40.1909 , J 9 ( 3 , 29 ) = 7.0711 , J 9 ( 3 , 30 ) = 40.1909 , J 9 ( 3 , 33 ) = 0.7071 ,
J 9 ( 3 , 35 ) = 40.1909 , J 9 ( 4 , 27 ) = 1 , J 9 ( 4 , 30 ) = 1 , J 9 ( 4 , 35 ) = 1 , J 9 ( 5 , 29 ) = 1 , J 9 ( 6 , 30 ) = 1.4141 ,
J 9 ( 6 , 34 ) = 1 , J 9 ( 7 , 31 ) = 1.
Δ OPL e 2 = Δ P 0 y 0 + Δ v 1 + 5 Δ ξ e 1 + 40 Δ ξ air + 1.5 Δ q e 1 + Δ v 2 5 Δ ω e 2 x 0 5 Δ ω e 2 z 0 + 17.071 Δ ξ e 2 + 2.21924 Δ m e 2 + 28.69239 Δ ψ e 2 + 13 Δ σ e 2 .
[ Δ P e 2 x 0 Δ P e 2 y 0 Δ P e 2 z 0 Δ e 2 x 0 Δ e 2 y 0 Δ e 2 z 0 ] = [ 0.0244 0 0 0.0216 0 0 ] Δ P 0 x 0 + [ 0 0.0173 0.0173 0 0.0153 0.0153 ] Δ P 0 z 0 + [ 0 32.122 32.122 0 0.5333 0.5333 ] Δ β 0 + [ 45.4274 0 0 0.7542 0 0 ] Δ α 0 + [ 0.9756 0 0 0.0216 0 0 ] Δ t e 1 x 0 + [ 0 1 0 0 0 0 ] ( Δ v 1 + Δ q e 1 + Δ v 2 ) + [ 0 0.6898 0.6898 0 0.0153 0.0153 ] Δ t e 1 z 0 + [ 0 24.2821 24.2821 0 0.563 0.563 ] Δ ω e 1 x 0 + [ 34.3401 0 0 0.7962 0 0 ] Δ ω e 1 z 0 + [ 0 4.2854 4.2854 0 0 0 ] Δ ω e 2 x 0 + [ 5 3.5355 1.4645 0.7071 0 0 ] Δ ω e 2 y 0 + [ 8.1316 5 0 0.2929 0 0 ] Δ ω e 2 z 0 + [ 0 0.7071 0.2929 0 0 0 ] Δ h e 2 + [ 0 1.2071 0.2071 0 0 0 ] Δ m e 2 + [ 0 29.1421 19.1421 0 2.0506 2.0506 ] Δ ψ e 2 + [ 0 10 10 0 1.8385 1.8385 ] Δ σ e 2 .
v = 1 v = 41 A 0 ε v d ε v = Dzrot ( ψ e 3 ) yrot ( 45 ° ) tran ( w e 2 , 0 , h e 3 ) xrot ( ψ e 2 + σ e 3 ) tran ( 0.5 w e 2 , v 1 q e 1 v 2 h e 2 2 gap glue , g e 2 ) d ψ e 3 + zrot ( ψ e 3 ) yrot ( 45 ° ) Dtran ( 1 , 0 , 0 ) xrot ( ψ e 2 + σ e 3 ) tran ( 0.5 w e 2 , v 1 q e 1 v 2 h e 2 2 gap glue , g e 2 ) d w e 2 zrot ( ψ e 3 ) yrot ( 45 ° ) Dtran ( 0 , 0 , 1 ) xrot ( ψ e 2 + σ e 3 ) tran ( 0.5 w e 2 , v 1 q e 1 v 2 h e 2 2 gap glue , g e 2 ) d h e 3 + zrot ( ψ e 3 ) yrot ( 45 ° ) tran ( w e 2 , 0 , h e 3 ) Dxrot ( ψ e 2 + σ e 3 ) tran ( 0.5 w e 2 , v 1 q e 1 v 2 h e 2 2 gap glue , g e 2 ) ( d σ e 3 d ψ e 2 ) zrot ( ψ e 3 ) yrot ( 45 ° ) tran ( w e 2 , 0 , h e 3 ) xrot ( ψ e 2 + σ e 3 ) Dyrot ( 0 ) tran ( 0.5 w e 2 , v 1 q e 1 v 2 h e 2 2 gap glue , g e 2 ) d ω 0 e 3 y zrot ( ψ e 3 ) yrot ( 45 ° ) tran ( w e 2 , 0 , h e 3 ) xrot ( ψ e 2 + σ e 3 ) Dzrot ( 0 ) tran ( 0.5 w e 2 , v 1 q e 1 v 2 h e 2 2 gap glue , g e 2 ) d ω 0 e 3 z + zrot ( ψ e 3 ) yrot ( 45 ° ) tran ( w e 2 , 0 , h e 3 ) xrot ( ψ e 2 + σ e 3 ) Dtran ( 0 , 0 , 1 ) d g e 2 zrot ( ψ e 3 ) yrot ( 45 ° ) tran ( w e 2 , 0 , h e 3 ) xrot ( ψ e 2 + σ e 3 ) Dtran ( 0 , 1 , 0 ) ( d v 1 + d q e 1 + d v 2 + d h e 2 + 2 gap glue ) 0.5 zrot ( ψ e 3 ) yrot ( 45 ° ) tran ( w e 2 , 0 , h e 3 ) xrot ( ψ e 2 + σ e 3 ) Dtran ( 0 , 0 , 1 ) d w e 2 .
tran ( t x , t y , t z ) tran ( q x , q y , q z ) = tran ( t x + q x , t y + q y , t z + q z ) ,
tran ( t x , t y , t z ) Dtran ( ε 1 , 0 , 0 ) tran ( q x , q y , q z ) = Dtran ( 1 , 0 , 0 ) ,
tran ( t x , t y , t z ) Dtran ( 0 , ε 1 , 0 ) tran ( q x , q y , q z ) = Dtran ( 0 , 1 , 0 ) ,
tran ( t x , t y , t z ) Dtran ( 0 , 0 , ε 1 ) tran ( q x , q y , q z ) = Dtran ( 0 , 0 , 1 ) .

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