Abstract

Based on the analysis of second-order moments, a generalized canonical representation of a two-dimensional optical signal is proposed, which is associated with the angular Poincaré sphere. Vortex-free (or zero-twist) optical beams arise on the equator of this sphere, while beams with a maximum vorticity (or maximum twist) are located at the poles. An easy way is shown how the latitude on the sphere, which is a measure for the degree of vorticity, can be derived from the second-order moments. The latitude is invariant when the beam propagates through a first-order optical system between conjugate planes. To change the vorticity of a beam, a system that does not operate between conjugate planes is needed, with the gyrator as the prime representative of such a system. A direct way is derived to find an optical system (consisting of a lens, a magnifier, a rotator, and a gyrator) that transforms a beam with an arbitrary moment matrix into its canonical form.

© 2010 Optical Society of America

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References

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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2009 (1)

2007 (1)

2005 (1)

2004 (1)

2003 (2)

2002 (1)

A. Ya. Bekshaev, M. V. Vasnetsov, V. G. Denisenko, and M. S. Soskin, “Transformation of the orbital angular momentum of a beam with optical vortex in a astigmatic optical system,” JETP Lett. 75, 127-130 (2002).
[CrossRef]

2001 (2)

2000 (2)

1999 (2)

1998 (2)

1997 (1)

M. S. Soskin, V. N. Gorshkov, M. V. Vasnetsov, J. T. Malos, and N. R. Heckenberg, “Topological charge and angular momentum of light beams carrying vortices,” Phys. Rev. A 56, 163-165 (1997).
[CrossRef]

1996 (1)

R. Simon, A. T. Friberg, and E. Wolf, “Transfer of radiance by twisted Gaussian Schell-model beams in paraxial systems,” J. Eur. Opt. Soc. A: Pure Appl. Opt. 5, 331-343 (1996).
[CrossRef]

1995 (1)

1994 (2)

A. T. Friberg, B. Tervonen, and J. Turunen, “Interpretation and experimental demonstration of twisted Gaussian Schell-model beams,” J. Opt. Soc. Am. A 11, 1818-1826 (1994).
[CrossRef]

D. Ambrosini, V. Bagini, F. Gori, and M. Santarsiero, “Twisted Gaussian Schell-model beams: a superposition model,” J. Mod. Opt. 41, 1391-1399 (1994).
[CrossRef]

1993 (3)

1991 (2)

1986 (1)

1936 (1)

J. Williamson, “On the algebraic problem concerning the normal forms of linear dynamical systems,” Am. J. Math. 58, 141-163 (1936).
[CrossRef]

Agarwal, G. S.

Alieva, T.

Ambrosini, D.

D. Ambrosini, V. Bagini, F. Gori, and M. Santarsiero, “Twisted Gaussian Schell-model beams: a superposition model,” J. Mod. Opt. 41, 1391-1399 (1994).
[CrossRef]

Bagini, V.

D. Ambrosini, V. Bagini, F. Gori, and M. Santarsiero, “Twisted Gaussian Schell-model beams: a superposition model,” J. Mod. Opt. 41, 1391-1399 (1994).
[CrossRef]

Bastiaans, M. J.

Bekshaev, A. Ya.

A. Ya. Bekshaev, M. S. Soskin, and M. V. Vasnetsov, “Optical vortex symmetry breakdown and decomposition of the orbital angular momentum of the light beams,” J. Opt. Soc. Am. A 20, 1635-1643 (2003).
[CrossRef]

A. Ya. Bekshaev, M. V. Vasnetsov, V. G. Denisenko, and M. S. Soskin, “Transformation of the orbital angular momentum of a beam with optical vortex in a astigmatic optical system,” JETP Lett. 75, 127-130 (2002).
[CrossRef]

Born, M.

M. Born and E. Wolf, Principles of Optics (Pergamon, 1980).

Calvo, G. F.

Courtial, J.

Denisenko, V. G.

A. Ya. Bekshaev, M. V. Vasnetsov, V. G. Denisenko, and M. S. Soskin, “Transformation of the orbital angular momentum of a beam with optical vortex in a astigmatic optical system,” JETP Lett. 75, 127-130 (2002).
[CrossRef]

Friberg, A. T.

R. Simon, A. T. Friberg, and E. Wolf, “Transfer of radiance by twisted Gaussian Schell-model beams in paraxial systems,” J. Eur. Opt. Soc. A: Pure Appl. Opt. 5, 331-343 (1996).
[CrossRef]

A. T. Friberg, B. Tervonen, and J. Turunen, “Interpretation and experimental demonstration of twisted Gaussian Schell-model beams,” J. Opt. Soc. Am. A 11, 1818-1826 (1994).
[CrossRef]

Gori, F.

D. Ambrosini, V. Bagini, F. Gori, and M. Santarsiero, “Twisted Gaussian Schell-model beams: a superposition model,” J. Mod. Opt. 41, 1391-1399 (1994).
[CrossRef]

Gorshkov, V. N.

M. S. Soskin, V. N. Gorshkov, M. V. Vasnetsov, J. T. Malos, and N. R. Heckenberg, “Topological charge and angular momentum of light beams carrying vortices,” Phys. Rev. A 56, 163-165 (1997).
[CrossRef]

Heckenberg, N. R.

M. S. Soskin, V. N. Gorshkov, M. V. Vasnetsov, J. T. Malos, and N. R. Heckenberg, “Topological charge and angular momentum of light beams carrying vortices,” Phys. Rev. A 56, 163-165 (1997).
[CrossRef]

Helgason, S.

S. Helgason, Differential Geometry, Lie Groups and Symmetric Species (Academic, 1978), Chap. VI.

Luneburg, R. K.

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

Malos, J. T.

M. S. Soskin, V. N. Gorshkov, M. V. Vasnetsov, J. T. Malos, and N. R. Heckenberg, “Topological charge and angular momentum of light beams carrying vortices,” Phys. Rev. A 56, 163-165 (1997).
[CrossRef]

Martínez-Herrero, R.

Mejías, P. M.

Movilla, J. M.

Mukunda, N.

Padgett, M. J.

Santarsiero, M.

D. Ambrosini, V. Bagini, F. Gori, and M. Santarsiero, “Twisted Gaussian Schell-model beams: a superposition model,” J. Mod. Opt. 41, 1391-1399 (1994).
[CrossRef]

Serna, J.

Simon, R.

Soskin, M. S.

A. Ya. Bekshaev, M. S. Soskin, and M. V. Vasnetsov, “Optical vortex symmetry breakdown and decomposition of the orbital angular momentum of the light beams,” J. Opt. Soc. Am. A 20, 1635-1643 (2003).
[CrossRef]

A. Ya. Bekshaev, M. V. Vasnetsov, V. G. Denisenko, and M. S. Soskin, “Transformation of the orbital angular momentum of a beam with optical vortex in a astigmatic optical system,” JETP Lett. 75, 127-130 (2002).
[CrossRef]

M. S. Soskin, M. V. Vasnetsov, “Singular optics,” Prog. Opt. 42, 219-276 (2001).
[CrossRef]

M. S. Soskin, V. N. Gorshkov, M. V. Vasnetsov, J. T. Malos, and N. R. Heckenberg, “Topological charge and angular momentum of light beams carrying vortices,” Phys. Rev. A 56, 163-165 (1997).
[CrossRef]

Sundar, K.

Tervonen, B.

Turunen, J.

Vasnetsov, M. V.

A. Ya. Bekshaev, M. S. Soskin, and M. V. Vasnetsov, “Optical vortex symmetry breakdown and decomposition of the orbital angular momentum of the light beams,” J. Opt. Soc. Am. A 20, 1635-1643 (2003).
[CrossRef]

A. Ya. Bekshaev, M. V. Vasnetsov, V. G. Denisenko, and M. S. Soskin, “Transformation of the orbital angular momentum of a beam with optical vortex in a astigmatic optical system,” JETP Lett. 75, 127-130 (2002).
[CrossRef]

M. S. Soskin, M. V. Vasnetsov, “Singular optics,” Prog. Opt. 42, 219-276 (2001).
[CrossRef]

M. S. Soskin, V. N. Gorshkov, M. V. Vasnetsov, J. T. Malos, and N. R. Heckenberg, “Topological charge and angular momentum of light beams carrying vortices,” Phys. Rev. A 56, 163-165 (1997).
[CrossRef]

Williamson, J.

J. Williamson, “On the algebraic problem concerning the normal forms of linear dynamical systems,” Am. J. Math. 58, 141-163 (1936).
[CrossRef]

Wolf, E.

R. Simon, A. T. Friberg, and E. Wolf, “Transfer of radiance by twisted Gaussian Schell-model beams in paraxial systems,” J. Eur. Opt. Soc. A: Pure Appl. Opt. 5, 331-343 (1996).
[CrossRef]

M. Born and E. Wolf, Principles of Optics (Pergamon, 1980).

Wolf, K. B.

Am. J. Math. (1)

J. Williamson, “On the algebraic problem concerning the normal forms of linear dynamical systems,” Am. J. Math. 58, 141-163 (1936).
[CrossRef]

J. Eur. Opt. Soc. A: Pure Appl. Opt. (1)

R. Simon, A. T. Friberg, and E. Wolf, “Transfer of radiance by twisted Gaussian Schell-model beams in paraxial systems,” J. Eur. Opt. Soc. A: Pure Appl. Opt. 5, 331-343 (1996).
[CrossRef]

J. Mod. Opt. (1)

D. Ambrosini, V. Bagini, F. Gori, and M. Santarsiero, “Twisted Gaussian Schell-model beams: a superposition model,” J. Mod. Opt. 41, 1391-1399 (1994).
[CrossRef]

J. Opt. Soc. Am. A (13)

J. Serna, R. Martínez-Herrero, and P. M. Mejías, “Parametric characterization of general partially coherent beams propagating through ABCD optical systems,” J. Opt. Soc. Am. A 8, 1094-1098 (1991).
[CrossRef]

K. Sundar, N. Mukunda, and R. Simon, “Coherent-mode decomposition of general anisotropic Gaussian Schell-model beams,” J. Opt. Soc. Am. A 12, 560-569 (1995).
[CrossRef]

R. Simon and N. Mukunda, “Twist phase in Gaussian-beam optics,” J. Opt. Soc. Am. A 15, 2373-2382 (1998).
[CrossRef]

M. J. Bastiaans, “Wigner distribution function applied to twisted Gaussian light propagating in first-order optical systems,” J. Opt. Soc. Am. A 17, 2475-2480 (2000).
[CrossRef]

A. Ya. Bekshaev, M. S. Soskin, and M. V. Vasnetsov, “Optical vortex symmetry breakdown and decomposition of the orbital angular momentum of the light beams,” J. Opt. Soc. Am. A 20, 1635-1643 (2003).
[CrossRef]

R. Simon and N. Mukunda, “Iwasawa decomposition in first-order optics: universal treatment of shape-invariant propagation for coherent and partially coherent beams,” J. Opt. Soc. Am. A 15, 2146-2155 (1998).
[CrossRef]

R. Simon and K. B. Wolf, “Structure of the set of paraxial optical systems,” J. Opt. Soc. Am. A 17, 342-355 (2000).
[CrossRef]

R. Simon and N. Mukunda, “Twisted Gaussian Schell-model beams,” J. Opt. Soc. Am. A 10, 95-109 (1993).
[CrossRef]

R. Simon, K. Sundar, and N. Mukunda, “Twisted Gaussian Schell-model beams. I. Symmetry structure and normal-mode spectrum,” J. Opt. Soc. Am. A 10, 2008-2016 (1993).
[CrossRef]

K. Sundar, R. Simon, and N. Mukunda, “Twisted Gaussian Schell-model beams. II. Spectrum analysis and propagation characteristics,” J. Opt. Soc. Am. A 10, 2017-2023 (1993).
[CrossRef]

A. T. Friberg, B. Tervonen, and J. Turunen, “Interpretation and experimental demonstration of twisted Gaussian Schell-model beams,” J. Opt. Soc. Am. A 11, 1818-1826 (1994).
[CrossRef]

G. S. Agarwal, “SU(2) structure of the Poincaré sphere for light beams with orbital angular momentum,” J. Opt. Soc. Am. A 16, 2914-2916 (1999).
[CrossRef]

M. J. Bastiaans, “Application of the Wigner distribution function to partially coherent light,” J. Opt. Soc. Am. A 3, 1227-1238 (1986).
[CrossRef]

JETP Lett. (1)

A. Ya. Bekshaev, M. V. Vasnetsov, V. G. Denisenko, and M. S. Soskin, “Transformation of the orbital angular momentum of a beam with optical vortex in a astigmatic optical system,” JETP Lett. 75, 127-130 (2002).
[CrossRef]

Opt. Lett. (7)

Optik (1)

M. J. Bastiaans, “Second-order moments of the Wigner distribution function in first-order optical systems,” Optik 88, 163-168 (1991).

Phys. Rev. A (1)

M. S. Soskin, V. N. Gorshkov, M. V. Vasnetsov, J. T. Malos, and N. R. Heckenberg, “Topological charge and angular momentum of light beams carrying vortices,” Phys. Rev. A 56, 163-165 (1997).
[CrossRef]

Prog. Opt. (1)

M. S. Soskin, M. V. Vasnetsov, “Singular optics,” Prog. Opt. 42, 219-276 (2001).
[CrossRef]

Other (4)

M. Born and E. Wolf, Principles of Optics (Pergamon, 1980).

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

S. Helgason, Differential Geometry, Lie Groups and Symmetric Species (Academic, 1978), Chap. VI.

K. B. Wolf, Geometric Optics on Phase Space (Springer, 2004).

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Figures (1)

Fig. 1
Fig. 1

Plot of Q 3 ( λ x λ y ) versus T ( λ x λ y ) for different values of the ratio r 2 = ( λ x λ y ) 2 4 λ x λ y : r = 5 (dashed curve), r = 2 (dashed–dotted curve), r = 1 (dotted curve), and r 0 (solid curve).

Equations (105)

Equations on this page are rendered with MathJax. Learn more.

W ( r , q ) = Γ ( r + 1 2 r , r 1 2 r ) exp ( i 2 π q t r ) d r .
M = 1 E [ r q ] [ r t , q t ] W ( r , q ) d r d q = 1 E [ x x x y x u x v x y y y y u y v x u y u u u u v x v y v u v v v ] × W ( x , y , u , v ) d x d y d u d v
M = [ m x x m x y m x u m x v m x y m y y m y u m y v m x u m y u m u u m u v m x v m y v m u v m v v ] [ M r r M r q M r q t M q q ] ,
[ r o q o ] = T [ r i q i ] = [ A B C D ] [ r i q i ] .
T = [ A B C D ] = [ D t B t C t A t ] 1 , or T 1 = J T t J with J = i [ 0 I I 0 ] = J 1 .
M o = [ M r r M r q M r q t M q q ] o = [ A B C D ] [ M r r M r q M r q t M q q ] i [ A B C D ] t = T M i T t .
M = T o Δ T o t ,
Δ = [ Λ 0 0 Λ ] with Λ = [ λ x 0 0 λ y ] and λ x λ y > 0 .
( M J ) 2 T o = T o Δ 2 .
λ x λ y = det M I 1 ,
λ x 2 + λ y 2 = ( m x x m u u m x u 2 ) + ( m y y m v v m y v 2 ) + 2 ( m x y m u v m x v m y u ) I 2 ,
( λ x ± λ y ) 2 = I 2 ± 2 I 1
λ x , y = 1 2 ( I 2 + 2 I 1 ± I 2 2 I 1 ) .
T o = [ A B C D ] = [ I 0 L I ] [ S 0 0 S 1 ] [ X Y Y X ] T l ( L ) T m ( S ) T o ( X + i Y ) ,
L = ( C A t + D B t ) ( A A t + B B t ) 1 = L t ,
S = ( A A t + B B t ) 1 2 = S t ,
U = X + i Y = ( A A t + B B t ) 1 2 ( A + i B ) = ( U ) 1 ,
U f ( γ 1 , γ 2 ) = [ exp ( i γ 1 ) 0 0 exp ( i γ 2 ) ] ,
U r ( ϕ ) = [ cos ϕ sin ϕ sin ϕ cos ϕ ] ,
U g ( ϕ ) = [ cos ϕ i sin ϕ i sin ϕ cos ϕ ] ,
O ( U ) = R ( α ) G ( β ) F ( γ x ψ , γ y + ψ ) ,
U = [ exp ( i γ x ) cos ϕ exp [ i ( γ y + γ ) ] sin ϕ exp [ i ( γ x γ ) ] sin ϕ exp ( i γ y ) cos ϕ ] = U f ( 1 2 γ , 1 2 γ ) U r ( ϕ ) U f ( 1 2 γ , 1 2 γ ) U f ( γ x , γ y ) = U r ( α ) U g ( β ) U f ( ψ , ψ ) U f ( γ x , γ y ) ,
sin 2 β = sin 2 ϕ sin γ ,
cos 2 ϕ = cos 2 α cos 2 β ,
M o = T o Δ T o t = [ X Y Y X ] [ Λ 0 0 Λ ] [ X t Y t Y t X t ] = [ X Λ X t + Y Λ Y t X Λ Y t + Y Λ X t X Λ Y t Y Λ X t X Λ X t + Y Λ Y t ] [ M r r o M r q o ( M r q o ) t M q q o ] = [ M r r o M r q o M r q o M r r o ] .
M r r o + i M r q o = U Λ U .
M r r o + i M r q o = U r ( α ) U g ( β ) Λ U g ( β ) U r ( α ) .
m x x o + m y y o = λ x + λ y ,
m x x o m y y o = ( λ x λ y ) cos 2 β cos 2 α ,
2 m x y o = ( λ x λ y ) cos 2 β sin 2 α ,
2 m x v o = ( λ x λ y ) sin 2 β ,
Q 1 = m x x o m y y o = Q cos 2 β cos 2 α ,
Q 2 = 2 m x y o = Q cos 2 β sin 2 α ,
Q 3 = 2 m x v o = Q sin 2 β ,
Q = Q 1 2 + Q 2 2 + Q 3 2 = λ x λ y ,
[ Q 1 , Q 2 , Q 3 ] = [ Q cos 2 β cos 2 α , Q cos 2 β sin 2 α , Q sin 2 β ] ,
M ¯ o ( β ) = [ M ¯ r r o ( β ) M ¯ r q o ( β ) M ¯ r q o ( β ) M ¯ r r o ( β ) ] ,
M ¯ r r o ( β ) + i M ¯ r q o ( β ) = U g ( β ) Λ U g ( β ) ,
m ¯ x x o = λ x cos 2 β + λ y sin 2 β ,
m ¯ y y o = λ x sin 2 β + λ y cos 2 β ,
2 m ¯ x v o = ( λ x λ y ) sin 2 β = 2 m x v o .
det M ¯ r r o ( β ) = [ 1 2 ( λ x λ y ) sin 2 β ] 2 + λ x λ y ,
M = [ M r r M r q M r q t M q q ] = [ I 0 L I ] M L [ I L 0 I ] ,
M L = [ S 0 0 S 1 ] [ U r ( α ) 0 0 U r ( α ) ] M ¯ o ( β ) × [ U r ( α ) 0 0 U r ( α ) ] [ S 0 0 S 1 ] [ M r r L M r q L ( M r q L ) t M q q L ] ,
M r r L = S U r ( α ) M ¯ r r o ( β ) U r ( α ) S = M r r ,
M q q L = S 1 U r ( α ) M ¯ r r o ( β ) U r ( α ) S 1 ,
M r q L = S U r ( α ) M ¯ r q o ( β ) U r ( α ) S 1 .
S 1 M r r S 1 = S M q q L S = U r ( α ) M ¯ r r o ( β ) U r ( α ) ,
det M r r = ( det S ) 2 det M ¯ r r o ( β ) .
M = [ m x x m x u m x u m u u ] = [ 1 0 l 1 ] [ s 0 0 s 1 ] [ λ 0 0 λ ] [ s 0 0 s 1 ] [ 1 l 0 1 ] = λ [ s 2 s 2 l s 2 l s 2 s 2 l 2 ] ,
M = [ M r r M r q M r q t M q q ] = [ I 0 L I ] [ S 0 0 S 1 ] [ λ I 0 0 λ I ] [ S 0 0 S 1 ] [ I L 0 I ] = λ [ S 2 S 2 L L S 2 S 2 L S 2 L ] ,
Z o ± = ( C + D Z i ± ) ( A + B Z i ± ) 1 .
M = [ M r r M r q M r q t M q q ] = [ I 0 M r q t M r r 1 I ] [ M r r 0 0 M q q M r q t M r r 1 M r q ] [ I M r r 1 M r q 0 I ] .
M r r 1 2 ( M r r 1 M r q ) M r r 1 2 = M r r 1 2 M r q M r r 1 2 = M r r 1 2 ( M r q M r r ) M r r 1 2 ,
T = ( m x u m y v ) m x y + m x v m y y m x x m y u ( det M r r ) 1 2 .
Λ = m x v m y u ,
Λ o = Λ ¯ o = 2 m x v o = ( λ x λ y ) sin 2 β ,
T o = T ¯ o = ( λ x + λ y ) m x v o ( m x v o ) 2 + λ x λ y .
T = [ A 0 L A ( A t ) 1 ] ,
( M r r ) o = A ( M r r ) i A t ,
( M r q ) o = A ( M r r ) i A t L + A ( M r q ) i A 1 ,
( M r q M r r ) o = A ( M r r ) i A t L A ( M r r ) i A t + A ( M r q M r r ) i A t .
( det M r r ) o 1 2 = ( det M r r ) i 1 2 det A .
Q 3 = ( λ x λ y ) sin 2 β = 2 T λ x λ y ( λ x + λ y ) 2 T 2 .
T = [ a U r ( α ) b U r ( α ) c U r ( α ) d U r ( α ) ] with a d b c = 1 .
M r r = m x x I , M q q = m u u I , M r q = [ m x u m x v m x v m x u ] ,
[ M r r 0 0 M q q M r q t M r r 1 M r q ] = T m ( S ) T r ( α ) G T r ( α ) T m ( S ) ,
M = T l ( M r q t M r r 1 ) T m ( S ) T r ( α ) G T r ( α ) T m ( S ) T l t ( M r q t M r r 1 ) ,
T l ( M r q t M r r 1 ) = [ I 0 M r q t M r r 1 I ] .
M r q t M r r 1 = L s + h σ with σ = [ 0 1 1 0 ] ,
L s = 1 2 ( M r q t M r r 1 + M r r 1 M r q ) ,
h σ = 1 2 ( M r q t M r r 1 M r r 1 M r q ) ,
M = T l ( L s ) T l ( h σ ) T m ( S ) T r ( α ) G T r ( α ) T m ( S ) T l t ( h σ ) T l t ( L s ) .
M = T l ( L s + L a ) T m ( m S ) T r ( α ) T g ( β ) Δ T g ( β ) T r ( α ) T m ( m S ) T l t ( L s + L a ) ,
L a = l S 1 [ sin 2 α cos 2 α cos 2 α sin 2 α ] S 1 ,
with l = g x g y g x + g y ( h det S ) = T 2 λ x λ y ( λ x λ y ) 2 T 2 ( λ x + λ y ) 2 T 2 ,
and m 4 = 1 + 4 g x g y ( g x + g y ) 2 ( h det S ) 2 = 1 + T 2 ( λ x + λ y ) 2 T 2 .
M L = [ M r r M r q L ( M r q L ) t M q q L ] = [ I 0 L I ] M [ I L 0 I ] .
S = [ s x x s x y s x y s y y ] = [ cos θ sin θ sin θ cos θ ] [ s x 0 0 s y ] [ cos θ sin θ sin θ cos θ ] ,
P = [ p x x p x y p x y p y y ] = [ cos θ sin θ sin θ cos θ ] [ s x 2 0 0 s y 2 ] [ cos θ sin θ sin θ cos θ ] ,
[ p y y p x y p x y p x x ] [ m x x m x y m x y m y y ] = ( det S ) 2 [ m u u L m u v L m u v L m v v L ] [ p x x p x y p x y p y y ] [ m u u m u v m u v m v v ] [ p x x p x y p x y p y y ] ,
[ m u u m x y + m u v m x x 0 m y y + m u u m x y + m u v m x y + m u v m x x + m v v 0 m y y m x y + m u v m v v ] [ p x x p x y p y y ] = 0 ;
[ m u u m x y + m u v m x x 0 m y y + m u u m x y + m u v m x y + m u v m x x + m v v 0 ] [ p x x p x y p y y ] = 0 .
p x x p x y = ( m x x + m v v ) ( m x y m u v ) ,
p y y p x y = ( m y y + m u u ) ( m x y m u v ) ,
p x y 2 [ ( m x x + m v v ) ( m y y + m u u ) ( m x y m u v ) 2 1 ] = det M r r det M ,
s x , y 2 = 1 2 ( p x x + p y y ) ± 1 2 ( p x x p y y ) 2 + p x y 2 ,
tan θ = ( s x 2 p x x ) p x y = p x y ( s x 2 p y y ) = p x y ( p x x s y 2 ) = ( p y y s y 2 ) p x y ,
p x x 2 = m x x m u u L = s x 4 and p y y 2 = m y y m v v L = s y 4 .
M = T l ( L s ) T l ( h σ ) T m ( S ) T r ( α ) G T r ( α ) T m ( S ) T l t ( h σ ) T l t ( L s ) .
T l ( h σ ) T m ( S ) T r ( α ) = T m ( S ) T r ( α ) T l ( h σ det S ) ,
M = T l ( L s ) T m ( S ) T r ( α ) T l ( h ̃ σ ) G T l t ( h ̃ σ ) T r ( α ) T m ( S ) T l t ( L s ) ,
[ g x 0 0 g x h ̃ 0 g y g y h ̃ 0 0 g y h ̃ g x + g y h ̃ 2 0 g x h ̃ 0 0 g y + g x h ̃ 2 ] .
T l ( L ) T m ( S ) T g ( β ) Δ T g ( β ) T m ( S ) T l t ( L ) ,
L = [ 0 l l 0 ] with l = g x g y g x + g y h ̃ ,
S = m I with m 4 = 1 + 4 g x g y ( g x + g y ) 2 h ̃ 2 ,
M = T l ( L s ) T m ( S ) T r ( α ) T l ( L ) T m ( m I ) T g ( β ) Δ × T g ( β ) T m ( m I ) T l t ( L ) T r ( α ) T m ( S ) T l t ( L s ) ,
M = T l ( L s + L a ) T m ( m S ) T r ( α ) T g ( β ) Δ T g ( β ) T r ( α ) T m ( m S ) T l t ( L s + L a ) ,
L a = S 1 U r ( α ) L U r ( α ) S 1 ,
( λ x ± λ y ) 2 = ( g x ± g y ) 2 + 4 g x g y h ̃ 2 ,
λ x λ y = g x g y ,
T = 2 h ̃ g x g y ,
l = T 2 λ x λ y ( λ x λ y ) 2 T 2 ( λ x + λ y ) 2 T 2
and m 4 = 1 + T 2 ( λ x + λ y ) 2 T 2 ,
sin 2 β = 4 g x g y h ̃ ( g x + g y ) ( g x g y ) 2 + 4 g x g y h ̃ 2 .

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