Abstract

The closed-form propagation expressions of an edge dislocation nested in a general elliptical Gaussian beam through aligned and misaligned paraxial optical ABCD systems are derived and used to study the dynamic evolution behavior of the edge dislocation propagating in free space and through a misaligned lens. It is shown that the noncanonical vortex and dynamic inversion of the topological charge, as well as a new edge dislocation may appear under certain conditions. The results are illustrated analytically and numerically.

© 2009 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. M. S. Soskin and M. V. Vasnetsov, “Singular optics,” in Progress in Optics, Vol. XLII, E.Wolf, ed. (North-Holland, 2001), pp. 219-276.
    [CrossRef]
  2. D. G. Grier, “A revolution in optical manipulation,” Nature (London) 424, 21-27 (2003).
  3. G. Indebetouw, “Optical vortices and their propagation,” J. Mod. Opt. 40, 73-87 (1993).
    [CrossRef]
  4. F. S. Roux, “Coupling of noncanonical optical vortices,” J. Opt. Soc. Am. B 21, 664-670 (2004).
    [CrossRef]
  5. Y. S. Kivshar, A. Nepomnyashchy, V. Tikhonenko, J. Christou, and B. Luther-Davies, “Vortex-stripe soliton interactions,” Opt. Lett. 25, 123-125 (2000).
    [CrossRef]
  6. D. V. Petrov, “Vortex-edge dislocation interaction in a linear medium,” Opt. Commun. 188, 307-312 (2001).
    [CrossRef]
  7. D. V. Petrov, “Splitting of an edge dislocation by an optical vortex,” Opt. Quantum Electron. 34, 759-773 (2002).
    [CrossRef]
  8. G. Molina-Terriza, J. Recolons, J. P. Torres, L. Torner, and E. M. Wright, “Observation of the dynamical inversion of topological charge of an optical vortex,” Phys. Rev. Lett. 87, 023902 (2001).
    [CrossRef]
  9. A. V. Mamaev, M. Saffman, and A. A. Zozulya, “Vortex evolution and bound pair formation in anisotropic nonlinear optical media,” Phys. Rev. Lett. 77, 4544-4547 (1996).
    [CrossRef] [PubMed]
  10. A. V. Mamaev, M. Saffman, and A. A. Zozulya, “Decay of high order optical vortices in anisotropic nonlinear optical media,” Phys. Rev. Lett. 78, 2108-2111 (1997).
    [CrossRef]
  11. V. Tikhonenko, J. Christou, B. Luther-Davies, and Y. S. Kivshar, “Observation of vortex solitons created by the instability of dark soliton stripes,” Opt. Lett. 21, 1129-1131 (1996).
    [CrossRef] [PubMed]
  12. Q. Lin, S. Wang, J. Alda, and E. Bernabeu, “Transformation of non-symmetric Gaussian beam into symmetric one by means of tensor ABCD law,” Optik (Stuttgart) 85, 67-72 (1990).
  13. Q. Lin and L. Wang, “Collins formula and tensor ABCD law in spatial-frequency domain,” Opt. Commun. 185, 263-269 (2000).
    [CrossRef]
  14. R. Simon, N. Mukunda, and E. C. G. Sudarshan, “Partially coherent beams and a generalized ABCD-law,” Opt. Commun. 65, 322-328 (1988).
    [CrossRef]
  15. A. Erdelyi, W. Magnus, and F. Oberhettinger, Tables of Integral Transforms (McGraw-Hill, 1954).
  16. Y. Cai and D. Ge, “Propagation of various dark hollow beams through an apertured paraxial ABCD optical system,” Phys. Lett. A 357, 72-80 (2006).
    [CrossRef]
  17. H. Weber, “Collins integral for misaligned optical elements,” J. Mod. Opt. 53, 2793-2801 (2006).
    [CrossRef]
  18. S. Wang and L. Ronchi, “Principles and design of optical arrays,” in Progress in Optics, Vol. XXV, E.Wolf, ed. (Elsevier Science, 1988), p. 279.

2006 (2)

Y. Cai and D. Ge, “Propagation of various dark hollow beams through an apertured paraxial ABCD optical system,” Phys. Lett. A 357, 72-80 (2006).
[CrossRef]

H. Weber, “Collins integral for misaligned optical elements,” J. Mod. Opt. 53, 2793-2801 (2006).
[CrossRef]

2004 (1)

2003 (1)

D. G. Grier, “A revolution in optical manipulation,” Nature (London) 424, 21-27 (2003).

2002 (1)

D. V. Petrov, “Splitting of an edge dislocation by an optical vortex,” Opt. Quantum Electron. 34, 759-773 (2002).
[CrossRef]

2001 (2)

G. Molina-Terriza, J. Recolons, J. P. Torres, L. Torner, and E. M. Wright, “Observation of the dynamical inversion of topological charge of an optical vortex,” Phys. Rev. Lett. 87, 023902 (2001).
[CrossRef]

D. V. Petrov, “Vortex-edge dislocation interaction in a linear medium,” Opt. Commun. 188, 307-312 (2001).
[CrossRef]

2000 (2)

Q. Lin and L. Wang, “Collins formula and tensor ABCD law in spatial-frequency domain,” Opt. Commun. 185, 263-269 (2000).
[CrossRef]

Y. S. Kivshar, A. Nepomnyashchy, V. Tikhonenko, J. Christou, and B. Luther-Davies, “Vortex-stripe soliton interactions,” Opt. Lett. 25, 123-125 (2000).
[CrossRef]

1997 (1)

A. V. Mamaev, M. Saffman, and A. A. Zozulya, “Decay of high order optical vortices in anisotropic nonlinear optical media,” Phys. Rev. Lett. 78, 2108-2111 (1997).
[CrossRef]

1996 (2)

A. V. Mamaev, M. Saffman, and A. A. Zozulya, “Vortex evolution and bound pair formation in anisotropic nonlinear optical media,” Phys. Rev. Lett. 77, 4544-4547 (1996).
[CrossRef] [PubMed]

V. Tikhonenko, J. Christou, B. Luther-Davies, and Y. S. Kivshar, “Observation of vortex solitons created by the instability of dark soliton stripes,” Opt. Lett. 21, 1129-1131 (1996).
[CrossRef] [PubMed]

1993 (1)

G. Indebetouw, “Optical vortices and their propagation,” J. Mod. Opt. 40, 73-87 (1993).
[CrossRef]

1990 (1)

Q. Lin, S. Wang, J. Alda, and E. Bernabeu, “Transformation of non-symmetric Gaussian beam into symmetric one by means of tensor ABCD law,” Optik (Stuttgart) 85, 67-72 (1990).

1988 (1)

R. Simon, N. Mukunda, and E. C. G. Sudarshan, “Partially coherent beams and a generalized ABCD-law,” Opt. Commun. 65, 322-328 (1988).
[CrossRef]

Alda, J.

Q. Lin, S. Wang, J. Alda, and E. Bernabeu, “Transformation of non-symmetric Gaussian beam into symmetric one by means of tensor ABCD law,” Optik (Stuttgart) 85, 67-72 (1990).

Bernabeu, E.

Q. Lin, S. Wang, J. Alda, and E. Bernabeu, “Transformation of non-symmetric Gaussian beam into symmetric one by means of tensor ABCD law,” Optik (Stuttgart) 85, 67-72 (1990).

Cai, Y.

Y. Cai and D. Ge, “Propagation of various dark hollow beams through an apertured paraxial ABCD optical system,” Phys. Lett. A 357, 72-80 (2006).
[CrossRef]

Christou, J.

Erdelyi, A.

A. Erdelyi, W. Magnus, and F. Oberhettinger, Tables of Integral Transforms (McGraw-Hill, 1954).

Ge, D.

Y. Cai and D. Ge, “Propagation of various dark hollow beams through an apertured paraxial ABCD optical system,” Phys. Lett. A 357, 72-80 (2006).
[CrossRef]

Grier, D. G.

D. G. Grier, “A revolution in optical manipulation,” Nature (London) 424, 21-27 (2003).

Indebetouw, G.

G. Indebetouw, “Optical vortices and their propagation,” J. Mod. Opt. 40, 73-87 (1993).
[CrossRef]

Kivshar, Y. S.

Lin, Q.

Q. Lin and L. Wang, “Collins formula and tensor ABCD law in spatial-frequency domain,” Opt. Commun. 185, 263-269 (2000).
[CrossRef]

Q. Lin, S. Wang, J. Alda, and E. Bernabeu, “Transformation of non-symmetric Gaussian beam into symmetric one by means of tensor ABCD law,” Optik (Stuttgart) 85, 67-72 (1990).

Luther-Davies, B.

Magnus, W.

A. Erdelyi, W. Magnus, and F. Oberhettinger, Tables of Integral Transforms (McGraw-Hill, 1954).

Mamaev, A. V.

A. V. Mamaev, M. Saffman, and A. A. Zozulya, “Decay of high order optical vortices in anisotropic nonlinear optical media,” Phys. Rev. Lett. 78, 2108-2111 (1997).
[CrossRef]

A. V. Mamaev, M. Saffman, and A. A. Zozulya, “Vortex evolution and bound pair formation in anisotropic nonlinear optical media,” Phys. Rev. Lett. 77, 4544-4547 (1996).
[CrossRef] [PubMed]

Molina-Terriza, G.

G. Molina-Terriza, J. Recolons, J. P. Torres, L. Torner, and E. M. Wright, “Observation of the dynamical inversion of topological charge of an optical vortex,” Phys. Rev. Lett. 87, 023902 (2001).
[CrossRef]

Mukunda, N.

R. Simon, N. Mukunda, and E. C. G. Sudarshan, “Partially coherent beams and a generalized ABCD-law,” Opt. Commun. 65, 322-328 (1988).
[CrossRef]

Nepomnyashchy, A.

Oberhettinger, F.

A. Erdelyi, W. Magnus, and F. Oberhettinger, Tables of Integral Transforms (McGraw-Hill, 1954).

Petrov, D. V.

D. V. Petrov, “Splitting of an edge dislocation by an optical vortex,” Opt. Quantum Electron. 34, 759-773 (2002).
[CrossRef]

D. V. Petrov, “Vortex-edge dislocation interaction in a linear medium,” Opt. Commun. 188, 307-312 (2001).
[CrossRef]

Recolons, J.

G. Molina-Terriza, J. Recolons, J. P. Torres, L. Torner, and E. M. Wright, “Observation of the dynamical inversion of topological charge of an optical vortex,” Phys. Rev. Lett. 87, 023902 (2001).
[CrossRef]

Ronchi, L.

S. Wang and L. Ronchi, “Principles and design of optical arrays,” in Progress in Optics, Vol. XXV, E.Wolf, ed. (Elsevier Science, 1988), p. 279.

Roux, F. S.

Saffman, M.

A. V. Mamaev, M. Saffman, and A. A. Zozulya, “Decay of high order optical vortices in anisotropic nonlinear optical media,” Phys. Rev. Lett. 78, 2108-2111 (1997).
[CrossRef]

A. V. Mamaev, M. Saffman, and A. A. Zozulya, “Vortex evolution and bound pair formation in anisotropic nonlinear optical media,” Phys. Rev. Lett. 77, 4544-4547 (1996).
[CrossRef] [PubMed]

Simon, R.

R. Simon, N. Mukunda, and E. C. G. Sudarshan, “Partially coherent beams and a generalized ABCD-law,” Opt. Commun. 65, 322-328 (1988).
[CrossRef]

Soskin, M. S.

M. S. Soskin and M. V. Vasnetsov, “Singular optics,” in Progress in Optics, Vol. XLII, E.Wolf, ed. (North-Holland, 2001), pp. 219-276.
[CrossRef]

Sudarshan, E. C. G.

R. Simon, N. Mukunda, and E. C. G. Sudarshan, “Partially coherent beams and a generalized ABCD-law,” Opt. Commun. 65, 322-328 (1988).
[CrossRef]

Tikhonenko, V.

Torner, L.

G. Molina-Terriza, J. Recolons, J. P. Torres, L. Torner, and E. M. Wright, “Observation of the dynamical inversion of topological charge of an optical vortex,” Phys. Rev. Lett. 87, 023902 (2001).
[CrossRef]

Torres, J. P.

G. Molina-Terriza, J. Recolons, J. P. Torres, L. Torner, and E. M. Wright, “Observation of the dynamical inversion of topological charge of an optical vortex,” Phys. Rev. Lett. 87, 023902 (2001).
[CrossRef]

Vasnetsov, M. V.

M. S. Soskin and M. V. Vasnetsov, “Singular optics,” in Progress in Optics, Vol. XLII, E.Wolf, ed. (North-Holland, 2001), pp. 219-276.
[CrossRef]

Wang, L.

Q. Lin and L. Wang, “Collins formula and tensor ABCD law in spatial-frequency domain,” Opt. Commun. 185, 263-269 (2000).
[CrossRef]

Wang, S.

Q. Lin, S. Wang, J. Alda, and E. Bernabeu, “Transformation of non-symmetric Gaussian beam into symmetric one by means of tensor ABCD law,” Optik (Stuttgart) 85, 67-72 (1990).

S. Wang and L. Ronchi, “Principles and design of optical arrays,” in Progress in Optics, Vol. XXV, E.Wolf, ed. (Elsevier Science, 1988), p. 279.

Weber, H.

H. Weber, “Collins integral for misaligned optical elements,” J. Mod. Opt. 53, 2793-2801 (2006).
[CrossRef]

Wright, E. M.

G. Molina-Terriza, J. Recolons, J. P. Torres, L. Torner, and E. M. Wright, “Observation of the dynamical inversion of topological charge of an optical vortex,” Phys. Rev. Lett. 87, 023902 (2001).
[CrossRef]

Zozulya, A. A.

A. V. Mamaev, M. Saffman, and A. A. Zozulya, “Decay of high order optical vortices in anisotropic nonlinear optical media,” Phys. Rev. Lett. 78, 2108-2111 (1997).
[CrossRef]

A. V. Mamaev, M. Saffman, and A. A. Zozulya, “Vortex evolution and bound pair formation in anisotropic nonlinear optical media,” Phys. Rev. Lett. 77, 4544-4547 (1996).
[CrossRef] [PubMed]

J. Mod. Opt. (2)

G. Indebetouw, “Optical vortices and their propagation,” J. Mod. Opt. 40, 73-87 (1993).
[CrossRef]

H. Weber, “Collins integral for misaligned optical elements,” J. Mod. Opt. 53, 2793-2801 (2006).
[CrossRef]

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

Nature (London) (1)

D. G. Grier, “A revolution in optical manipulation,” Nature (London) 424, 21-27 (2003).

Opt. Commun. (3)

Q. Lin and L. Wang, “Collins formula and tensor ABCD law in spatial-frequency domain,” Opt. Commun. 185, 263-269 (2000).
[CrossRef]

R. Simon, N. Mukunda, and E. C. G. Sudarshan, “Partially coherent beams and a generalized ABCD-law,” Opt. Commun. 65, 322-328 (1988).
[CrossRef]

D. V. Petrov, “Vortex-edge dislocation interaction in a linear medium,” Opt. Commun. 188, 307-312 (2001).
[CrossRef]

Opt. Lett. (2)

Opt. Quantum Electron. (1)

D. V. Petrov, “Splitting of an edge dislocation by an optical vortex,” Opt. Quantum Electron. 34, 759-773 (2002).
[CrossRef]

Optik (Stuttgart) (1)

Q. Lin, S. Wang, J. Alda, and E. Bernabeu, “Transformation of non-symmetric Gaussian beam into symmetric one by means of tensor ABCD law,” Optik (Stuttgart) 85, 67-72 (1990).

Phys. Lett. A (1)

Y. Cai and D. Ge, “Propagation of various dark hollow beams through an apertured paraxial ABCD optical system,” Phys. Lett. A 357, 72-80 (2006).
[CrossRef]

Phys. Rev. Lett. (3)

G. Molina-Terriza, J. Recolons, J. P. Torres, L. Torner, and E. M. Wright, “Observation of the dynamical inversion of topological charge of an optical vortex,” Phys. Rev. Lett. 87, 023902 (2001).
[CrossRef]

A. V. Mamaev, M. Saffman, and A. A. Zozulya, “Vortex evolution and bound pair formation in anisotropic nonlinear optical media,” Phys. Rev. Lett. 77, 4544-4547 (1996).
[CrossRef] [PubMed]

A. V. Mamaev, M. Saffman, and A. A. Zozulya, “Decay of high order optical vortices in anisotropic nonlinear optical media,” Phys. Rev. Lett. 78, 2108-2111 (1997).
[CrossRef]

Other (3)

A. Erdelyi, W. Magnus, and F. Oberhettinger, Tables of Integral Transforms (McGraw-Hill, 1954).

M. S. Soskin and M. V. Vasnetsov, “Singular optics,” in Progress in Optics, Vol. XLII, E.Wolf, ed. (North-Holland, 2001), pp. 219-276.
[CrossRef]

S. Wang and L. Ronchi, “Principles and design of optical arrays,” in Progress in Optics, Vol. XXV, E.Wolf, ed. (Elsevier Science, 1988), p. 279.

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (7)

Fig. 1
Fig. 1

(a) α r versus z z R in free-space propagation. (b) α r versus p in free-space propagation. (c) α r versus w 0 x in free-space propagation. Refer to text for calculation parameters.

Fig. 2
Fig. 2

(a) Contour lines of phase for c = 0 corresponding to z z R = 1 in Fig. 1a. (b) Contour lines of phase for c = 0 corresponding to p = 0.1 in Fig. 1b. (c) Contour lines of phase for c = 0 corresponding to p = p c 2 in Fig. 1b. (d) Contour lines of phase for c = 0 corresponding to w 0 x = w 0 x c in Fig. 1c. (e) Contour lines of phase for c = 0 corresponding to w 0 x = 4 mm in Fig. 1c.

Fig. 3
Fig. 3

(a) 3D trajectory of a vortex for c = 0.5 mm in free-space propagation. (b) 3D trajectory of a vortex for c = 0 in free-space propagation.

Fig. 4
Fig. 4

(a) Position of a vortex versus p in the x direction in free-space propagation. (b) Position of a vortex versus p in the y direction in free-space propagation.

Fig. 5
Fig. 5

α f r versus z f in propagation through a misaligned lens.

Fig. 6
Fig. 6

Contour lines of phase for c = 0 corresponding to z f = 1 in Fig. 5.

Fig. 7
Fig. 7

3D trajectory of a vortex for c = 0.5 mm in propagation through a misaligned lens.

Equations (46)

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

E ( x 1 , y 1 , 0 ) = ( p x 1 y 1 + c ) exp ( i k 2 r 1 T Q 1 1 r 1 ) ,
Q 1 1 = [ 2 i k w 0 x 2 2 i k w 0 x y 2 2 i k w 0 x y 2 2 i k w 0 y 2 ] ,
E ( x , y , z ) = i k 2 π [ det ( B ) ] 1 2 E ( x 1 , y 1 , 0 ) exp [ i k 2 ( r 1 T B 1 A r 1 ) ] × exp [ i k 2 ( 2 r 1 T B 1 r + r T D B 1 r ) ] d r 1 ,
exp [ ( x b ) 2 2 a ] H n ( x ) d x = 2 π a ( 1 2 a ) n 2 H n ( b 1 2 a ) ,
E ( x , y , z ) = i k 2 π [ det ( A + B Q 1 1 ) ] 1 exp [ i k 2 r T Q 2 1 r ] F ,
F = p V 1 V 2 + c ,
V = ( V 1 V 2 ) = ( A + B Q 1 1 ) 1 r ,
Q 2 1 = ( C + D Q 1 1 ) ( A + B Q 1 1 ) 1 ( tensor A B C D law ) [ 14 ] .
A = [ 1 0 0 1 ] , B = [ z 0 0 z ] , C = [ 0 0 0 0 ] , D = [ 1 0 0 1 ] .
F = i k w 0 2 ( p x y ) i k w 0 2 + 2 z + c .
F = q 0 ( x + i α y + β ) ,
q 0 = p ( 1 2 i z k w 0 y 2 ) i 2 z k w 0 x y 2 1 2 i z k w 0 x 2 + 4 z 2 k 2 w 0 x y 4 2 i z k w 0 y 2 4 z 2 k 2 w 0 x 2 w 0 y 2 ,
α = α r + i α i ( noncanonical strength [ 8 ] ) ,
β = β r + i β i ,
α r = 2 z k p ( p w 0 x y 2 + 1 w 0 x 2 ) ( p w 0 y 2 + 1 w 0 x y 2 ) p 2 + 4 z 2 k 2 ( p w 0 y 2 + 1 w 0 x y 2 ) 2 ,
α i = 4 z 2 k 2 ( p w 0 y 2 + 1 w 0 x y 2 ) ( p w 0 x y 2 + 1 w 0 x 2 ) + p p 2 + 4 z 2 k 2 ( p w 0 y 2 + 1 w 0 x y 2 ) 2 ;
β r = c p ( 1 4 z 2 k 2 w 0 x 2 w 0 y 2 + 4 z 2 k 2 w 0 x y 4 ) + 4 z 2 k 2 ( 1 w 0 x 2 + 1 w 0 y 2 ) ( p w 0 y 2 + 1 w 0 x y 2 ) p 2 + 4 z 2 k 2 ( p w 0 y 2 + 1 w 0 x y 2 ) 2 ,
β i = 2 c z k ( 1 4 z 2 k 2 w 0 x 2 w 0 y 2 + 4 z 2 k 2 w 0 x y 4 ) ( p w 0 y 2 + 1 w 0 x y 2 ) p ( 1 w 0 x 2 + 1 w 0 y 2 ) p 2 + 4 z 2 k 2 ( p w 0 y 2 + 1 w 0 x y 2 ) 2 .
p c 1 , 2 = w 0 x y 2 2 w 0 y 2 w 0 x y 2 2 w 0 x 2 ± 1 + ( w 0 x y 2 2 w 0 y 2 w 0 x y 2 2 w 0 x 2 ) 2
w 0 x c = p w 0 y 2 w 0 x y 2 w 0 x y 2 p + w 0 y 2 ( 1 p 2 )
E ( x , y , z ) = i k 2 π [ det ( B ) ] 1 2 E ( x 1 , y 1 , 0 ) exp [ i k 2 ( r 1 T B 1 A r 1 2 r 1 T B 1 r + r T D B 1 r ) ] × exp [ i k 2 ( r 1 T B 1 e f + r T B 1 g h ) ] d r 1 ,
e f T = ( e f ) , g h T = ( g h ) ,
A = [ a 0 0 a ] , B = [ b 0 0 b ] , C = [ c 0 0 c ] , D = [ d 0 0 d ] ,
e = 2 ( α T ε x + β T ε x ) ,
f = 2 ( α T ε y + β T ε y ) ,
g = 2 ( b γ T d α T ) ε x + 2 ( b δ T d β T ) ε x ,
h = 2 ( b γ T d α T ) ε y + 2 ( b δ T d β T ) ε y ,
α T = 1 a , β T = z b , γ T = c , δ T = ± 1 d .
A = [ 1 z f 0 0 1 z f ] , B = [ z 0 0 z ] ,
C = [ 1 f 0 0 1 f ] , D = [ 1 0 0 1 ] ,
α T = z f , β T = 0 , γ T = 1 f , δ T = 0 ,
e = 2 z ε x f , f = 2 z ε y f , g = 0 , h = 0 .
E ( x , y , z ) = i k 2 π [ det ( A + B Q 1 1 ) ] 1 exp [ i k 2 r T Q 2 1 r i k 2 r T B 1 g h ] × exp [ i k 2 r T B 1 T ( A + B Q 1 1 ) 1 e f ] exp [ i k 8 e f T B 1 T ( A + B Q 1 1 ) 1 e f ] F f ,
F f = p V f 1 V f 2 + c ,
V f = ( V f 1 V f 2 ) = ( A + B Q 1 1 ) 1 ( r e f 2 ) .
F f = p ( x z ε x f ) ( y z ε y f ) ( 1 z f 2 i z k w 0 2 ) + c .
F f = q 1 [ x z ε x f + i α f ( y z ε y f ) + β f ] ,
q 1 = p ( 1 z f 2 i z k w 0 y 2 ) i 2 z k w 0 x y 2 4 z 2 k 2 w 0 x y 4 + ( 1 z f 2 i z k w 0 x 2 ) ( 1 z f 2 i z k w 0 y 2 ) ,
α f = α f r + i α f i ,
β f = β f r + i β f i ,
α f r = ( 1 z f ) 2 z k p ( p w 0 x y 2 + 1 w 0 x 2 ) ( p w 0 y 2 + 1 w 0 x y 2 ) [ p ( 1 z f ) ] 2 + 4 z 2 k 2 ( p w 0 y 2 + 1 w 0 x y 2 ) 2 ,
α f i = 4 z 2 k 2 ( p w 0 x y 2 + 1 w 0 x 2 ) ( p w 0 y 2 + 1 w 0 x y 2 ) + p ( 1 z f ) 2 [ p ( 1 z f ) ] 2 + 4 z 2 k 2 ( p w 0 y 2 + 1 w 0 x y 2 ) 2 ;
β f r = c ( 1 z f ) p [ ( 1 z f ) 2 4 z 2 k 2 w 0 x 2 w 0 y 2 + 4 z 2 k 2 w 0 x y 4 ] + 4 z 2 k 2 ( 1 w 0 x 2 + 1 w 0 y 2 ) ( p w 0 y 2 + 1 w 0 x y 2 ) [ p ( 1 z f ) ] 2 + 4 z 2 k 2 ( p w 0 y 2 + 1 w 0 x y 2 ) 2 ,
β f i = 2 c z k ( p w 0 y 2 + 1 w 0 x y 2 ) [ ( 1 z f ) 2 4 z 2 k 2 w 0 x 2 w 0 y 2 + 4 z 2 k 2 w 0 x y 4 ] p ( 1 w 0 x 2 + 1 w 0 y 2 ) ( 1 z f ) 2 [ p ( 1 z f ) ] 2 + 4 z 2 k 2 ( p w 0 y 2 + 1 w 0 x y 2 ) 2 .
p c 1 , 2 = w 0 x y 2 2 w 0 y 2 w 0 x y 2 2 w 0 x 2 ± 1 + ( w 0 x y 2 2 w 0 y 2 w 0 x y 2 2 w 0 x 2 ) 2
w 0 x c = p w 0 y 2 w 0 x y 2 w 0 x y 2 p + w 0 y 2 ( 1 p 2 )

Metrics