Abstract

The statistical properties of the spatial derivatives of the Stokes parameters for a random polarization field are studied. Based on the Gaussian assumption for the electric fields, the six-dimensional joint probability density function for the derivatives of the Stokes parameters is obtained from the statistics of the derivatives of the random polarization field. Subsequently, three two-dimensional probability density functions of derivatives of each Stokes parameter and the corresponding six marginal probability density functions are given. Finally, the joint and marginal density functions of the magnitude of the gradient of Stokes parameters are also derived for the first time, to our knowledge.

© 2010 Optical Society of America

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References

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  1. J. W. Goodman, “Statistical properties of laser speckle patterns,” in Laser Speckle and Related Phenomena, J.C.Dainty, ed. (Springer-Verlag, 1975).
    [CrossRef]
  2. J. W. Goodman, Speckle Phenomena in Optics: Theory and Application (Roberts and Company, 2006).
  3. J. W. Goodman, Statistical Optics (Wiley, 2000).
  4. A. F. Fercher and P. F. Steeger, “First-order statistics of Stokes parameters in speckle fields,” Opt. Acta 28, 443–448 (1981).
    [CrossRef]
  5. D. Eliyahu, “Vector statistics of correlated Gaussian fields,” Phys. Rev. E 47, 2881–2892 (1993).
    [CrossRef]
  6. P. F. Steeger, T. Asakura, K. Zocha, and A. F. Fercher, “Statistics of the Stokes parameters in speckle fields,” J. Opt. Soc. Am. A 1, 677–682 (1984).
    [CrossRef]
  7. P. F. Steeger, “Probability density function of the intensity in partially polarized speckle fields,” Opt. Lett. 8, 528–530 (1983).
    [CrossRef] [PubMed]
  8. R. Barakat, “Statistics of the Stokes parameters,” J. Opt. Soc. Am. A 4, 1256–1263 (1987).
    [CrossRef]
  9. M. V. Berry and M. R. Dennis, “Polarization singularities in isotropic random vector waves,” Proc. R. Soc. London, Ser. A 457, 141–155 (2001).
    [CrossRef]
  10. M. R. Dennis, “Polarization singularities in paraxial vector fields: morphology and statistics,” Opt. Commun. 213, 201–221 (2002).
    [CrossRef]
  11. G. V. Bogatyryova, K. V. Felde, P. V. Polyanskii, and M. S. Soskin, “Nongeneric polarization singularities in combined vortex beams,” Opt. Spectrosc. 97, 782–789 (2004).
    [CrossRef]
  12. J. F. Nye, “Local solutions for the interaction of wave dislocations,” J. Opt. A, Pure Appl. Opt. 6, S251–S254 (2004).
    [CrossRef]
  13. J. F. Nye, “Lines of circular polarization in electromagnetic wave fields,” Proc. R. Soc. London, Ser. A 389, 279–290 (1983).
    [CrossRef]
  14. J. F. Nye and J. V. Hajnal, “The wave structure of monochromatic electromagnetic radiation,” Proc. R. Soc. London, Ser. A 409, 21–36 (1987).
    [CrossRef]
  15. J. F. Nye, Natural Focusing and the Fine Structure of Light (IOP, 1999).
  16. Y. Ohtsuka and K. Oka, “Contour mapping of the spatiotemporal state of polarization of light,” Appl. Opt. 33, 2633–2636 (1994).
    [CrossRef] [PubMed]
  17. K. J. Ebeling, “Statistical properties of spatial derivatives of the amplitude and intensity of monochromatic speckle patterns,” Opt. Acta 26, 1505–1521 (1979).
    [CrossRef]
  18. R. Barakat, “Zero-crossing rate of differentiated speckle intensity,” J. Opt. Soc. Am. A 11, 671–673 (1994).
    [CrossRef]
  19. D. A. Kessler and I. Freund, “Level-crossing densities in random wave fields,” J. Opt. Soc. Am. A 15, 1608–1618 (1998).
    [CrossRef]
  20. J. Ohtsubo, “Exact solution of the zero crossing rate of a differentiated speckle pattern,” Opt. Commun. 42, 13–18 (1982).
    [CrossRef]
  21. M. S. Longuet-Higgins, “The statistical analysis of a random moving surface,” Philos. Trans. R. Soc. London, Ser. A 249, 321–387 (1957).
    [CrossRef]
  22. E. Ochoa and J. W. Goodman, “Statistical properties of ray directions in a monochromatic speckle pattern,” J. Opt. Soc. Am. 73, 943–949 (1983).
    [CrossRef]
  23. M. Born and E. Wolf, Principle of Optics, 7th ed. (Cambridge Univ. Press, 1999).
  24. A. Papoulis, Probability, Random Variables, and Stochastic Processes (McGraw-Hill, 1960).
  25. I. I. Gradshteyn and I. M. Ryzhik, Table of Integrals Series and Products (Academic, 1965).
  26. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (Dover, 1965).

2004

G. V. Bogatyryova, K. V. Felde, P. V. Polyanskii, and M. S. Soskin, “Nongeneric polarization singularities in combined vortex beams,” Opt. Spectrosc. 97, 782–789 (2004).
[CrossRef]

J. F. Nye, “Local solutions for the interaction of wave dislocations,” J. Opt. A, Pure Appl. Opt. 6, S251–S254 (2004).
[CrossRef]

2002

M. R. Dennis, “Polarization singularities in paraxial vector fields: morphology and statistics,” Opt. Commun. 213, 201–221 (2002).
[CrossRef]

2001

M. V. Berry and M. R. Dennis, “Polarization singularities in isotropic random vector waves,” Proc. R. Soc. London, Ser. A 457, 141–155 (2001).
[CrossRef]

1998

1994

1993

D. Eliyahu, “Vector statistics of correlated Gaussian fields,” Phys. Rev. E 47, 2881–2892 (1993).
[CrossRef]

1987

J. F. Nye and J. V. Hajnal, “The wave structure of monochromatic electromagnetic radiation,” Proc. R. Soc. London, Ser. A 409, 21–36 (1987).
[CrossRef]

R. Barakat, “Statistics of the Stokes parameters,” J. Opt. Soc. Am. A 4, 1256–1263 (1987).
[CrossRef]

1984

1983

1982

J. Ohtsubo, “Exact solution of the zero crossing rate of a differentiated speckle pattern,” Opt. Commun. 42, 13–18 (1982).
[CrossRef]

1981

A. F. Fercher and P. F. Steeger, “First-order statistics of Stokes parameters in speckle fields,” Opt. Acta 28, 443–448 (1981).
[CrossRef]

1979

K. J. Ebeling, “Statistical properties of spatial derivatives of the amplitude and intensity of monochromatic speckle patterns,” Opt. Acta 26, 1505–1521 (1979).
[CrossRef]

1957

M. S. Longuet-Higgins, “The statistical analysis of a random moving surface,” Philos. Trans. R. Soc. London, Ser. A 249, 321–387 (1957).
[CrossRef]

Abramowitz, M.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (Dover, 1965).

Asakura, T.

Barakat, R.

Berry, M. V.

M. V. Berry and M. R. Dennis, “Polarization singularities in isotropic random vector waves,” Proc. R. Soc. London, Ser. A 457, 141–155 (2001).
[CrossRef]

Bogatyryova, G. V.

G. V. Bogatyryova, K. V. Felde, P. V. Polyanskii, and M. S. Soskin, “Nongeneric polarization singularities in combined vortex beams,” Opt. Spectrosc. 97, 782–789 (2004).
[CrossRef]

Born, M.

M. Born and E. Wolf, Principle of Optics, 7th ed. (Cambridge Univ. Press, 1999).

Dennis, M. R.

M. R. Dennis, “Polarization singularities in paraxial vector fields: morphology and statistics,” Opt. Commun. 213, 201–221 (2002).
[CrossRef]

M. V. Berry and M. R. Dennis, “Polarization singularities in isotropic random vector waves,” Proc. R. Soc. London, Ser. A 457, 141–155 (2001).
[CrossRef]

Ebeling, K. J.

K. J. Ebeling, “Statistical properties of spatial derivatives of the amplitude and intensity of monochromatic speckle patterns,” Opt. Acta 26, 1505–1521 (1979).
[CrossRef]

Eliyahu, D.

D. Eliyahu, “Vector statistics of correlated Gaussian fields,” Phys. Rev. E 47, 2881–2892 (1993).
[CrossRef]

Felde, K. V.

G. V. Bogatyryova, K. V. Felde, P. V. Polyanskii, and M. S. Soskin, “Nongeneric polarization singularities in combined vortex beams,” Opt. Spectrosc. 97, 782–789 (2004).
[CrossRef]

Fercher, A. F.

P. F. Steeger, T. Asakura, K. Zocha, and A. F. Fercher, “Statistics of the Stokes parameters in speckle fields,” J. Opt. Soc. Am. A 1, 677–682 (1984).
[CrossRef]

A. F. Fercher and P. F. Steeger, “First-order statistics of Stokes parameters in speckle fields,” Opt. Acta 28, 443–448 (1981).
[CrossRef]

Freund, I.

Goodman, J. W.

E. Ochoa and J. W. Goodman, “Statistical properties of ray directions in a monochromatic speckle pattern,” J. Opt. Soc. Am. 73, 943–949 (1983).
[CrossRef]

J. W. Goodman, Speckle Phenomena in Optics: Theory and Application (Roberts and Company, 2006).

J. W. Goodman, “Statistical properties of laser speckle patterns,” in Laser Speckle and Related Phenomena, J.C.Dainty, ed. (Springer-Verlag, 1975).
[CrossRef]

J. W. Goodman, Statistical Optics (Wiley, 2000).

Gradshteyn, I. I.

I. I. Gradshteyn and I. M. Ryzhik, Table of Integrals Series and Products (Academic, 1965).

Hajnal, J. V.

J. F. Nye and J. V. Hajnal, “The wave structure of monochromatic electromagnetic radiation,” Proc. R. Soc. London, Ser. A 409, 21–36 (1987).
[CrossRef]

Kessler, D. A.

Longuet-Higgins, M. S.

M. S. Longuet-Higgins, “The statistical analysis of a random moving surface,” Philos. Trans. R. Soc. London, Ser. A 249, 321–387 (1957).
[CrossRef]

Nye, J. F.

J. F. Nye, “Local solutions for the interaction of wave dislocations,” J. Opt. A, Pure Appl. Opt. 6, S251–S254 (2004).
[CrossRef]

J. F. Nye and J. V. Hajnal, “The wave structure of monochromatic electromagnetic radiation,” Proc. R. Soc. London, Ser. A 409, 21–36 (1987).
[CrossRef]

J. F. Nye, “Lines of circular polarization in electromagnetic wave fields,” Proc. R. Soc. London, Ser. A 389, 279–290 (1983).
[CrossRef]

J. F. Nye, Natural Focusing and the Fine Structure of Light (IOP, 1999).

Ochoa, E.

Ohtsubo, J.

J. Ohtsubo, “Exact solution of the zero crossing rate of a differentiated speckle pattern,” Opt. Commun. 42, 13–18 (1982).
[CrossRef]

Ohtsuka, Y.

Oka, K.

Papoulis, A.

A. Papoulis, Probability, Random Variables, and Stochastic Processes (McGraw-Hill, 1960).

Polyanskii, P. V.

G. V. Bogatyryova, K. V. Felde, P. V. Polyanskii, and M. S. Soskin, “Nongeneric polarization singularities in combined vortex beams,” Opt. Spectrosc. 97, 782–789 (2004).
[CrossRef]

Ryzhik, I. M.

I. I. Gradshteyn and I. M. Ryzhik, Table of Integrals Series and Products (Academic, 1965).

Soskin, M. S.

G. V. Bogatyryova, K. V. Felde, P. V. Polyanskii, and M. S. Soskin, “Nongeneric polarization singularities in combined vortex beams,” Opt. Spectrosc. 97, 782–789 (2004).
[CrossRef]

Steeger, P. F.

Stegun, I. A.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (Dover, 1965).

Wolf, E.

M. Born and E. Wolf, Principle of Optics, 7th ed. (Cambridge Univ. Press, 1999).

Zocha, K.

Appl. Opt.

J. Opt. A, Pure Appl. Opt.

J. F. Nye, “Local solutions for the interaction of wave dislocations,” J. Opt. A, Pure Appl. Opt. 6, S251–S254 (2004).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Opt. Acta

K. J. Ebeling, “Statistical properties of spatial derivatives of the amplitude and intensity of monochromatic speckle patterns,” Opt. Acta 26, 1505–1521 (1979).
[CrossRef]

A. F. Fercher and P. F. Steeger, “First-order statistics of Stokes parameters in speckle fields,” Opt. Acta 28, 443–448 (1981).
[CrossRef]

Opt. Commun.

M. R. Dennis, “Polarization singularities in paraxial vector fields: morphology and statistics,” Opt. Commun. 213, 201–221 (2002).
[CrossRef]

J. Ohtsubo, “Exact solution of the zero crossing rate of a differentiated speckle pattern,” Opt. Commun. 42, 13–18 (1982).
[CrossRef]

Opt. Lett.

Opt. Spectrosc.

G. V. Bogatyryova, K. V. Felde, P. V. Polyanskii, and M. S. Soskin, “Nongeneric polarization singularities in combined vortex beams,” Opt. Spectrosc. 97, 782–789 (2004).
[CrossRef]

Philos. Trans. R. Soc. London, Ser. A

M. S. Longuet-Higgins, “The statistical analysis of a random moving surface,” Philos. Trans. R. Soc. London, Ser. A 249, 321–387 (1957).
[CrossRef]

Phys. Rev. E

D. Eliyahu, “Vector statistics of correlated Gaussian fields,” Phys. Rev. E 47, 2881–2892 (1993).
[CrossRef]

Proc. R. Soc. London, Ser. A

J. F. Nye, “Lines of circular polarization in electromagnetic wave fields,” Proc. R. Soc. London, Ser. A 389, 279–290 (1983).
[CrossRef]

J. F. Nye and J. V. Hajnal, “The wave structure of monochromatic electromagnetic radiation,” Proc. R. Soc. London, Ser. A 409, 21–36 (1987).
[CrossRef]

M. V. Berry and M. R. Dennis, “Polarization singularities in isotropic random vector waves,” Proc. R. Soc. London, Ser. A 457, 141–155 (2001).
[CrossRef]

Other

J. F. Nye, Natural Focusing and the Fine Structure of Light (IOP, 1999).

J. W. Goodman, “Statistical properties of laser speckle patterns,” in Laser Speckle and Related Phenomena, J.C.Dainty, ed. (Springer-Verlag, 1975).
[CrossRef]

J. W. Goodman, Speckle Phenomena in Optics: Theory and Application (Roberts and Company, 2006).

J. W. Goodman, Statistical Optics (Wiley, 2000).

M. Born and E. Wolf, Principle of Optics, 7th ed. (Cambridge Univ. Press, 1999).

A. Papoulis, Probability, Random Variables, and Stochastic Processes (McGraw-Hill, 1960).

I. I. Gradshteyn and I. M. Ryzhik, Table of Integrals Series and Products (Academic, 1965).

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (Dover, 1965).

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Figures (2)

Fig. 1
Fig. 1

Normalized p.d.f. of derivatives of Stokes parameters: (a) joint density function P ( x S i , y S i ) ( i = 1 , 2 , 3 ) and (b) marginal density P ( μ S i ) ( μ = x , y ; i = 1 , 2 , 3 ) .

Fig. 2
Fig. 2

P.d.f. of the magnitude of the gradient for each Stokes parameter P ( | S i | ) ( i = 1 , 2 , 3 ) .

Equations (49)

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E ̃ η = 1 N k = 1 N a η k exp ( i ϕ η k )     ( η = x , y ) ,
S 0 = ( E x r ) 2 + ( E x i ) 2 + ( E y r ) 2 + ( E y i ) 2 = I x + I y ,
S 1 = ( E x r ) 2 + ( E x i ) 2 ( E y r ) 2 ( E y i ) 2 = I x I y ,
S 2 = 2 ( E x r E y r + E x i E y i ) = 2 I x I y   cos ( θ x θ y ) ,
S 3 = 2 ( E x r E y i E x i E y r ) = 2 I x I y   sin ( θ x θ y ) ,
P ( E x r , E x i , E y r , E y i , x E x r , x E x i , y E x r , y E x i , x E y r , x E y i , y E y r , y E y i ) = ( 2 6 π 6 σ 4 b x 2 b y 2 ) 1 exp ( Ψ 1 / 2 ) exp ( Ψ 2 / 2 ) ,
Ψ 1 = σ 2 [ ( E x r ) 2 + ( E x i ) 2 ] + b x 2 [ ( x E x r ) 2 + ( x E x i ) 2 ] + b x 2 [ ( y E x r ) 2 + ( y E x i ) 2 ] ,
Ψ 2 = σ 2 [ ( E y r ) 2 + ( E y i ) 2 ] + b y 2 [ ( x E y r ) 2 + ( x E y i ) 2 ] + b y 2 [ ( y E y r ) 2 + ( y E y i ) 2 ] .
E x r = I x   cos   θ x ,
E x i = I x   sin   θ x ,
x E x r = ( 2 I x ) 1 x I x   cos   θ x I x   sin   θ x x θ x ,
x E x i = ( 2 I x ) 1 x I x   sin   θ x + I x   cos   θ x x θ x ,
y E x r = ( 2 I x ) 1 y I x   cos   θ x I x   sin   θ x y θ x ,
y E x i = ( 2 I x ) 1 y I x   sin   θ x + I x   cos   θ x y θ x .
P ( I x , θ x , I y , θ y , x I x , x θ x , x I y , x θ y , y I x , y θ x , y I y , y θ y ) = ( 2 12 π 6 σ 4 b x 2 b y 2 ) 1 exp ( Τ ) ,
T = I x [ b x b y + σ 2 b y ( x θ x ) 2 + σ 2 b x ( y θ x ) 2 ] / ( 2 σ 2 b x b y ) + I y [ b x b y + σ 2 b y ( x θ y ) 2 + σ 2 b x ( y θ y ) 2 ] / ( 2 σ 2 b x b y ) + [ b y ( x I x ) 2 + b x ( y I x ) 2 ] / ( 8 I x b x b y ) + [ b y ( x I y ) 2 + b x ( y I y ) 2 ] / ( 8 I y b x b y ) .
ψ = θ x + θ y ,
ϕ = θ x θ y ,
x ψ = x θ x + x θ y ,
y ψ = y θ x + y θ y ,
x ϕ = x θ x x θ y ,
y ϕ = y θ x y θ y ,
P ( I x , ψ , I y , ϕ , x I x , x ψ , x I y , x ϕ , y I x , y ψ , y I y , y ϕ ) = f ( ψ , ϕ ) ( 2 15 π 6 σ 4 b x 2 b y 2 ) 1 exp ( K ) ,
f ( ψ , ϕ ) = { 1 , ψ + ϕ < 2 π , ψ ϕ > 2 π , ϕ > 0 1 , ψ ϕ < 2 π , ψ + ϕ > 2 π , ϕ < 0 0 , otherwise , }
K = ( I x + I y ) ( x ψ ) 2 / ( 8 b x ) + ( I x + I y ) ( y ψ ) 2 / ( 8 b y ) + ( I x I y ) ( b x 1 x ψ x ϕ + b y 1 y ψ y ϕ ) / 4 + ( I x + I y ) / ( 2 σ 2 ) + ( I x + I y ) ( x ϕ ) 2 / ( 8 b y ) + ( I x + I y ) ( y ϕ ) 2 / ( 8 b x ) + [ b y ( x I x ) 2 + b x ( y I x ) 2 ] / ( 8 I x b x b y ) + [ b y ( x I y ) 2 + b x ( y I y ) 2 ] / ( 8 I y b x b y ) .
P ( I x , I y , ϕ , x I x , x I y , x ϕ , y I x , y I y , y ϕ ) = ( 2 15 π 6 σ 4 b x 2 b y 2 ) 1 2 π 2 π + + exp ( K ) f ( ψ , ϕ ) d φ d ( x ψ ) d ( y ψ ) = g ( ϕ ) [ 2 12 π 6 σ 4 b 3 ( I x + I y ) ] 1 exp { I x I y [ ( ( x ϕ ) 2 + ( y ϕ ) 2 ) ] / [ 2 b ( I x + I y ) ] } exp [ ( I x + I y ) / ( 2 σ 2 ) ( x I x ) 2 / ( 8 b I x ) ( x I y ) 2 / ( 8 b I y ) ] exp [ ( y I x ) 2 / ( 8 b I x ) ( y I y ) 2 / ( 8 b I y ) ] / ( 8 I y b x b y ) ,
g ( ϕ ) = { | 4 π 2 ϕ | , for   2 π < ϕ < 2 π 0 , otherwise . }
S 1 = I x I y ,
S 2 = 2 I x I y   cos   ϕ ,
S 3 = 2 I x I y   sin   ϕ ,
x S 1 = x I x x I y ,
x S 2 = I y / I x x I x   cos   ϕ + I x / I y x I y   cos   ϕ 2 I x I y x ϕ   sin   ϕ ,
x S 3 = I y / I x x I x   sin   ϕ + I x / I y x I y   sin   ϕ + 2 I x I y x ϕ   cos   ϕ ,
y S 1 = y I x y I y ,
y S 2 = I y / I x y I x   cos   ϕ + I x / I y y I y   cos   ϕ 2 I x I y y ϕ   sin   ϕ ,
y S 3 = I y / I x y I x   sin   ϕ + I x / I y y I y   sin   ϕ + 2 I x I y y ϕ   cos   ϕ .
P ( S 1 , S 2 , S 3 , x S 1 , x S 2 , x S 3 , y S 1 , y S 2 , y S 3 ) = g ( arctan   S 3 / S 2 ) / [ 2 15 π 6 σ 4 b 3 ( S 1 2 + S 2 2 + S 3 2 ) 2 ] exp { S 1 2 + S 2 2 + S 3 2 / ( 2 σ 2 ) i = 1 3 [ ( x S i ) 2 + ( y S i ) 2 ] / ( 8 b S 1 2 + S 2 2 + S 3 2 ) } .
P ( S 0 , x S 1 , x S 2 , x S 3 , y S 1 , y S 2 , y S 3 ) = 1 2 11 π 3 σ 4 b 3 1 S 0 2 exp { S 0 2 σ 2 1 8 b S 0 i = 1 3 [ ( x S i ) 2 + ( y S i ) 2 ] } .
P ( x S 1 , x S 2 , x S 3 , y S 1 , y S 2 , y S 3 ) = ( 2 9 π 3 σ 5 b 5 / 2 ) 1 K 1 { i = 1 3 [ ( x S i ) 2 + ( y S i ) 2 ] / 2 b σ } { i = 1 3 [ ( x S i ) 2 + ( y S i ) 2 ] } 1 / 2 .
0 + t α 1   exp [ t x 2 / ( 4 t ) ] d t = K α ( x ) ( | x | / 2 ) α ,
P ( x S i , y S i ) = 1 32 π σ 3 b 3 / 2 ( x S i ) 2 + ( y S i ) 2 K 1 [ ( x S i ) 2 + ( y S i ) 2 / 2 b σ ]     ( i = 1 , 2 , 3 ) ,
P ( k S i ) = 1 4 σ 2 b π K 3 / 2 ( | k S i | / 2 σ b ) ( | k S i | / 2 σ b ) 3 / 2 = 1 16 σ 2 b exp ( | k S i | / 2 σ b ) ( | k S i | + 2 σ b )
( k = x , y ; i = 1 , 2 , 3 ) ,
| S i | = ( x S i ) 2 + ( y S i ) 2 ,
γ i = arctan ( y S i / x S i ) ,
P ( S 0 , | S 1 | , | S 2 | , | S 3 | ) = 1 2 8 σ 4 b 3 | S 1 S 2 S 3 | S 0 2 exp [ S 0 2 σ 2 1 8 b S 0 i = 1 3 | S i | 2 ] .
P ( | S 1 | , | S 2 | , | S 3 | ) = 1 2 6 σ 5 b 5 / 2 | S 1 S 2 S 3 | | S 1 | 2 + | S 2 | 2 + | S 3 | 2 K 1 ( | S 1 | 2 + | S 2 | 2 + | S 3 | 2 ) .
P ( S 0 , | S i | ) = 1 2 4 σ 4 b | S i | exp [ S 0 2 σ 2 | S i | 2 8 b S 0 ]     ( i = 1 , 2 , 3 ) .
P ( | S i | ) = 1 2 4 σ 3 b 3 / 2 | S i | 2 K 1 ( | S i | 2 b σ )     ( i = 1 , 2 , 3 ) .

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