Abstract

Optical aberration effects up to the second moment of Gaussian laser speckle are theoretically investigated for both partially and fully developed speckle. In the development, a plane-wave illuminated diffuser generates a phase-perturbed random field in the object plane that creates speckle in the image plane. Theoretical derivations show that image field statistics are generally non-circular Gaussian due to aberrations. Speckle statistics are not affected by odd-functional aberrations, such as coma, and dependency of aberrations is asymptotically ignorable for very weak or strong diffusers. Furthermore, Gaussian speckle contrast as a functional of optical aberrations exhibits a stationary point for the aberration free condition, where apparently contrast does not achieve a local maximum. Calculations of speckle contrast for several aberration conditions are also presented.

© 2009 Optical Society of America

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References

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  1. E. M. Gullikson, "Scattering from normal incidence EUV optics," Proc. SPIE 3331, 72-80 (1998).
    [CrossRef]
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    [CrossRef]
  3. N. A. Beaudry and T. D. Milster, "Effects of mask roughness and condenser scattering in EUVL systems," Proc. SPIE 3676, 653-662 (1999).
    [CrossRef]
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    [CrossRef] [PubMed]
  12. R. N. Singh and A. K. Singhal, "Formation of laser speckles under extra-axial aberrations," Opt. Quantum Electron. 12, 519-524 (1980).
    [CrossRef]
  13. A. Kumar and K. Singh, "Elongated laser speckles in imaging of a rough object with slit shaped illuminated region: Effect of off-axis aberrations," Optik 96, 115-119 (1994).
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    [CrossRef]
  15. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1996).
  16. H. H. Barrett and K. J. Myers, Foundations of Image Science (Wiley Series, 2004).
  17. J. Krim, I. Heyvaert, C. Van Haesendonck, and Y. Bruynseraede, "Scanning tunneling microscopy observation of self-affine fractal roughness in ion-bombarded film surfaces," Phys. Rev. Lett. 70, 57-60 (1993).
    [CrossRef] [PubMed]
  18. G. Palasantzas and J. Krim, "Effect of the form of the height-height correlation function on diffuse x-ray scattering from a self-affine surface," Phys. Rev. B 48, 2873-2877 (1993).
    [CrossRef]
  19. G. Palasantzas, "Roughness spectrum and surface width of self-affine fractal surfaces via the K-correlation model," Phys. Rev. B 48, 14472-14478 (1993).
    [CrossRef]
  20. I. R. Reed, "On a moment theorem for complex Gaussian processes," Trans. Inform. Theory IT-8, 194-195 (1962).
    [CrossRef]
  21. F. D. Neeser and J. L. Massey, "Proper complex random processes with applications to information theory," IEEE Trans. Inform. Theory 39, 1293-1302 (1993).
    [CrossRef]
  22. Bochner and Salomon, Lectures on Fourier integrals (Princeton University, 1959).

2004 (1)

1999 (1)

N. A. Beaudry and T. D. Milster, "Effects of mask roughness and condenser scattering in EUVL systems," Proc. SPIE 3676, 653-662 (1999).
[CrossRef]

1998 (2)

E. M. Gullikson, "Scattering from normal incidence EUV optics," Proc. SPIE 3331, 72-80 (1998).
[CrossRef]

N. A. Beaudry and T. D. Milster, "Scattering and Coherence in EUVL," Proc. SPIE 3331, 537-543 (1998).
[CrossRef]

1994 (1)

A. Kumar and K. Singh, "Elongated laser speckles in imaging of a rough object with slit shaped illuminated region: Effect of off-axis aberrations," Optik 96, 115-119 (1994).

1993 (4)

J. Krim, I. Heyvaert, C. Van Haesendonck, and Y. Bruynseraede, "Scanning tunneling microscopy observation of self-affine fractal roughness in ion-bombarded film surfaces," Phys. Rev. Lett. 70, 57-60 (1993).
[CrossRef] [PubMed]

G. Palasantzas and J. Krim, "Effect of the form of the height-height correlation function on diffuse x-ray scattering from a self-affine surface," Phys. Rev. B 48, 2873-2877 (1993).
[CrossRef]

G. Palasantzas, "Roughness spectrum and surface width of self-affine fractal surfaces via the K-correlation model," Phys. Rev. B 48, 14472-14478 (1993).
[CrossRef]

F. D. Neeser and J. L. Massey, "Proper complex random processes with applications to information theory," IEEE Trans. Inform. Theory 39, 1293-1302 (1993).
[CrossRef]

1986 (1)

1980 (2)

R. D. Bahuguna, K. K. Gupta, and K. Singh, "Speckle patterns of weak diffusers: effect of spherical aberration," Appl. Opt. 19, 1874-1878 (1980).
[CrossRef] [PubMed]

R. N. Singh and A. K. Singhal, "Formation of laser speckles under extra-axial aberrations," Opt. Quantum Electron. 12, 519-524 (1980).
[CrossRef]

1979 (1)

1977 (1)

1976 (1)

1975 (1)

J. W. Goodman, "Dependence of image speckle contrast on surface roughness," Opt. Commun. 14, 324-327 (1975).
[CrossRef]

1962 (1)

I. R. Reed, "On a moment theorem for complex Gaussian processes," Trans. Inform. Theory IT-8, 194-195 (1962).
[CrossRef]

Allebach, J. P.

Bahuguna, R. D.

Beaudry, N. A.

N. A. Beaudry and T. D. Milster, "Effects of mask roughness and condenser scattering in EUVL systems," Proc. SPIE 3676, 653-662 (1999).
[CrossRef]

N. A. Beaudry and T. D. Milster, "Scattering and Coherence in EUVL," Proc. SPIE 3331, 537-543 (1998).
[CrossRef]

Bruynseraede, Y.

J. Krim, I. Heyvaert, C. Van Haesendonck, and Y. Bruynseraede, "Scanning tunneling microscopy observation of self-affine fractal roughness in ion-bombarded film surfaces," Phys. Rev. Lett. 70, 57-60 (1993).
[CrossRef] [PubMed]

Gallagher, N. C.

Goodman, J. W.

J. W. Goodman, "Dependence of image speckle contrast on surface roughness," Opt. Commun. 14, 324-327 (1975).
[CrossRef]

Gullikson, E. M.

E. M. Gullikson, "Scattering from normal incidence EUV optics," Proc. SPIE 3331, 72-80 (1998).
[CrossRef]

Gupta, K. K.

Heyvaert, I.

J. Krim, I. Heyvaert, C. Van Haesendonck, and Y. Bruynseraede, "Scanning tunneling microscopy observation of self-affine fractal roughness in ion-bombarded film surfaces," Phys. Rev. Lett. 70, 57-60 (1993).
[CrossRef] [PubMed]

Krim, J.

J. Krim, I. Heyvaert, C. Van Haesendonck, and Y. Bruynseraede, "Scanning tunneling microscopy observation of self-affine fractal roughness in ion-bombarded film surfaces," Phys. Rev. Lett. 70, 57-60 (1993).
[CrossRef] [PubMed]

G. Palasantzas and J. Krim, "Effect of the form of the height-height correlation function on diffuse x-ray scattering from a self-affine surface," Phys. Rev. B 48, 2873-2877 (1993).
[CrossRef]

Kumar, A.

A. Kumar and K. Singh, "Elongated laser speckles in imaging of a rough object with slit shaped illuminated region: Effect of off-axis aberrations," Optik 96, 115-119 (1994).

Massey, J. L.

F. D. Neeser and J. L. Massey, "Proper complex random processes with applications to information theory," IEEE Trans. Inform. Theory 39, 1293-1302 (1993).
[CrossRef]

Milster, T. D.

N. A. Beaudry and T. D. Milster, "Effects of mask roughness and condenser scattering in EUVL systems," Proc. SPIE 3676, 653-662 (1999).
[CrossRef]

N. A. Beaudry and T. D. Milster, "Scattering and Coherence in EUVL," Proc. SPIE 3331, 537-543 (1998).
[CrossRef]

Murphy, P. K.

Naulleau, P. P.

Neeser, F. D.

F. D. Neeser and J. L. Massey, "Proper complex random processes with applications to information theory," IEEE Trans. Inform. Theory 39, 1293-1302 (1993).
[CrossRef]

Palasantzas, G.

G. Palasantzas, "Roughness spectrum and surface width of self-affine fractal surfaces via the K-correlation model," Phys. Rev. B 48, 14472-14478 (1993).
[CrossRef]

G. Palasantzas and J. Krim, "Effect of the form of the height-height correlation function on diffuse x-ray scattering from a self-affine surface," Phys. Rev. B 48, 2873-2877 (1993).
[CrossRef]

Reed, I. R.

I. R. Reed, "On a moment theorem for complex Gaussian processes," Trans. Inform. Theory IT-8, 194-195 (1962).
[CrossRef]

Singh, K.

Singh, R. N.

R. N. Singh and A. K. Singhal, "Formation of laser speckles under extra-axial aberrations," Opt. Quantum Electron. 12, 519-524 (1980).
[CrossRef]

Singhal, A. K.

R. N. Singh and A. K. Singhal, "Formation of laser speckles under extra-axial aberrations," Opt. Quantum Electron. 12, 519-524 (1980).
[CrossRef]

Stetson, K. A.

Van Haesendonck, C.

J. Krim, I. Heyvaert, C. Van Haesendonck, and Y. Bruynseraede, "Scanning tunneling microscopy observation of self-affine fractal roughness in ion-bombarded film surfaces," Phys. Rev. Lett. 70, 57-60 (1993).
[CrossRef] [PubMed]

Appl. Opt. (2)

IEEE Trans. Inform. Theory (1)

F. D. Neeser and J. L. Massey, "Proper complex random processes with applications to information theory," IEEE Trans. Inform. Theory 39, 1293-1302 (1993).
[CrossRef]

J. Opt. Soc. Am. (3)

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

Opt. Commun. (1)

J. W. Goodman, "Dependence of image speckle contrast on surface roughness," Opt. Commun. 14, 324-327 (1975).
[CrossRef]

Opt. Quantum Electron. (1)

R. N. Singh and A. K. Singhal, "Formation of laser speckles under extra-axial aberrations," Opt. Quantum Electron. 12, 519-524 (1980).
[CrossRef]

Optik (1)

A. Kumar and K. Singh, "Elongated laser speckles in imaging of a rough object with slit shaped illuminated region: Effect of off-axis aberrations," Optik 96, 115-119 (1994).

Phys. Rev. B (2)

G. Palasantzas and J. Krim, "Effect of the form of the height-height correlation function on diffuse x-ray scattering from a self-affine surface," Phys. Rev. B 48, 2873-2877 (1993).
[CrossRef]

G. Palasantzas, "Roughness spectrum and surface width of self-affine fractal surfaces via the K-correlation model," Phys. Rev. B 48, 14472-14478 (1993).
[CrossRef]

Phys. Rev. Lett. (1)

J. Krim, I. Heyvaert, C. Van Haesendonck, and Y. Bruynseraede, "Scanning tunneling microscopy observation of self-affine fractal roughness in ion-bombarded film surfaces," Phys. Rev. Lett. 70, 57-60 (1993).
[CrossRef] [PubMed]

Proc. SPIE (3)

E. M. Gullikson, "Scattering from normal incidence EUV optics," Proc. SPIE 3331, 72-80 (1998).
[CrossRef]

N. A. Beaudry and T. D. Milster, "Scattering and Coherence in EUVL," Proc. SPIE 3331, 537-543 (1998).
[CrossRef]

N. A. Beaudry and T. D. Milster, "Effects of mask roughness and condenser scattering in EUVL systems," Proc. SPIE 3676, 653-662 (1999).
[CrossRef]

Trans. Inform. Theory (1)

I. R. Reed, "On a moment theorem for complex Gaussian processes," Trans. Inform. Theory IT-8, 194-195 (1962).
[CrossRef]

Other (5)

Bochner and Salomon, Lectures on Fourier integrals (Princeton University, 1959).

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1996).

H. H. Barrett and K. J. Myers, Foundations of Image Science (Wiley Series, 2004).

J. C. Dainty,  et al. Laser speckle and related phenomena (Springer, 1984).

J. W. Goodman, Speckle phenomena in optics, theory and applications (Robert, 2007)

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Figures (3)

Fig. 1.
Fig. 1.

Conceptual layout for generating speckle in the image plane. A plane wave illuminates rough surface located in the object plane. The solid circles indicate contribution areas of the random object field for generating speckle at image points.

Fig. 2.
Fig. 2.

Speckle contrasts (a) and cross second moments (b) between real and imaginary fields at fixed observation point with spherical and defocus aberrations from -0.5λ to 0.5λ. Speckle contrast (c) and cross second moment (d) at normalized image fields from 0 to 1 with 0.5λ spherical, -0.1λ field curvature and -0.2λ astigmatism. All aberrations are of third order. Dotted lines in (b) and (d) indicate the combination of aberrations for a zero cross second moment. Speckle contrasts in (a) and (c) along these lines show relatively minimum values.

Fig. 3.
Fig. 3.

Aberrations when the cross second moments between real and imaginary fields are almost zero. Coordinates are a spatial frequency ξ = x xp /λr′ [um]-1 . Parameters in (a) and (b) are the same as Figs. 2-(b) and (d), respectively. Defocus and spherical aberrations for (a) are -0.175λ and 0.4λ, respectively. Normalized field locations Hx and Hy for (b) are 0.23 and 1, respectively. Units are wavelength of 13.5nm.

Equations (100)

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g R = g I = 0 , g R 2 = g I 2 , and g R g I = 0 ,
g ( x i ) = 1 m T f ( x o m T ) h coh ( x i x o ) d x o ,
f ( x o ) = exp [ i 2 πl ( x o ) ]
h coh ( x i x o ) = T ( m T ξ ) exp [ i 2 πW ( ξ ) ] exp [ i 2 π ( x i x o ) · ξ ] d ξ .
f R ( x o ) = exp [ 2 π 2 K l ( 0 ) ] = m R
f I ( x o ) = 0
f R ( x o 1 ) f R * ( x o 2 ) = R RR o ( Δ x o ) = m R 2 2 [ R α ( Δ x o ) + R β ( Δ x o ) ]
f I ( x o 1 ) f I * ( x o 2 ) = R II o ( Δ x o ) = m R 2 2 [ R α ( Δ x o ) R β ( Δ x o ) ]
f R ( x o 1 ) f I * ( x o 2 ) = f I ( x o 1 ) f R * ( x o 2 ) = 0 ,
R α ( Δ x o ) = exp { 4 π 2 K l ( Δ x o ) }
R β ( Δ x o ) = exp { 4 π 2 K l ( Δ x o ) } .
K l ( Δ x o ) = σ 2 exp [ ( Δ x o / L cor ) 2 H ] ,
I ( x i ) = C diff m T 2 f ( x o 1 m T ) f * ( x o 2 m T ) h coh ( x i x o 1 ) h coh * ( x i x o 2 ) d x o 1 d x o 2 ,
K s ( Δ x i ) = g ( Δ x i + x t ) 2 g ( x i ) 2 g ( x i ) 2 2 ,
K s ( Δ x i ) = 2 { [ R RR i ( Δ x i ) ] 2 + [ R II i ( Δ x i ) ] 2 + [ R RI i ( Δ x i ) ] 2 + [ R IR i ( Δ x i ) ] 2 [ c R 2 + c I 2 ] 2 }
C s = 2 [ R RR i ( 0 ) 2 + R II i ( 0 ) 2 + R RI i ( 0 ) 2 + R RI i ( 0 ) 2 ( c R 2 + c I 2 ) 2 ] 1 / 2 R RR i ( 0 ) + R II i ( 0 ) + i ( R RI i ( 0 ) R RI i ( 0 ) ) .
g R ( x i ) = 1 m T [ h coh , R ( x i x o ) f R ( x o m T ) h coh , I ( x i x o ) f I ( x o m T ) ] d x o
g I ( x i ) = 1 m T [ h coh , R ( x i x o ) f I ( x o m T ) + h coh , I ( x i x o ) f R ( x o m T ) ] d x o ,
h coh , R ( x i x o ) = T ( m T ξ ) exp [ i 2 π W o ( ξ ) ] cos [ 2 π W e ( ξ ) ] exp [ i 2 π ( x i x o ) · ξ ] d ξ
h coh , I ( x i x o ) = T ( m T ξ ) exp [ i 2 π W o ( ξ ) ] sin [ 2 π W e ( ξ ) ] exp [ i 2 π ( x i x o ) · ξ ] d ξ .
c R = m R m T cos [ 2 π W e ( 0 ) ]
c I = m R m T sin [ 2 π W e ( 0 ) ]
R RR i ( Δ x i ) = χ ind ( Δ x i ) + χ ab ( Δ x i )
R II i ( Δ x i ) = χ ind ( Δ x i ) χ ab ( Δ x i ) ,
χ ind ( Δ x i ) = m R 2 2 m T 2 F Δ x i { T ( m T ξ ) F ξ 1 [ R α ( Δ x o m T ) ] }
χ ab ( Δ x i ) = m R 2 2 m T 2 F Δ x i { T ( m T ξ ) cos [ 4 π W e ( ξ ) ] F ξ 1 [ R β ( Δ x o m T ) ] } .
R RI i ( Δ x i ) = m R 2 2 m T 2 F Δ x i { T ( m T ξ ) sin [ 4 π W e ( ξ ) ] F ξ 1 [ R β ( Δ x i m T ) ] } .
K s ( Δ x i ) = K s ind ( Δ x i ) + K s ab ( Δ x i ) ,
K s ind ( Δ x i ) = 4 [ χ ind ( Δ x i ) ] 2 2 ( c R 2 + c I 2 ) 2 ,
K s ab ( Δ x i ) = 4 [ χ ab ( Δ x i ) ] 2 + 4 [ ( R RI i Δ x i ) ] 2 ,
χ ab ( Δ x i ) m R 2 2 m T 2 cos [ 4 π W e ( 0 ) ]
R RI i ( Δ x i ) m R 2 2 m T 2 sin [ 4 π W e ( 0 ) ] .
χ ab ( Δ x i ) W e ( 0 ) = 4 π R RI i ( Δ x i ) , R RI i ( Δ x i ) W e ( 0 ) = 4 π χ ab ( Δ x i ) ,
Ψ l ( s ) = exp [ i 2 π s l ]
= exp [ 2 π 2 ( s , K l , s ) ] ,
f R ( r ) = 1 2 [ exp [ i 2 πl ( r ) ] + exp [ i 2 πl ( r ) ] ]
f I ( r ) = 1 2 i [ exp [ i 2 πl ( r ) ] exp [ i 2 πl ( r ) ] ] ,
exp [ i 2 πl ( r ) ] = exp [ i 2 π δ ( r r ) l ( r ) d r ] = Ψ l [ δ ( r r ) ] .
Ψ l [ δ ( r r ) ] = exp [ 2 π 2 δ ( r r ) K l ( r , r ) δ ( r r ) d r d r ]
= exp [ 2 π 2 K l r r ] ,
f R ( r ) = exp [ 2 π 2 K l r r ]
f R ( r ) = 0 .
R RR o r r
= 1 4 { exp [ i 2 πl ( r ) ] + exp [ i 2 πl ( r ) ] } { exp [ i 2 πl ( r ) ] + exp [ i 2 πl ( r ) ] }
= 1 4 exp [ i 2 πl ( r ) ] exp [ i 2 πl ( r ) ] + exp [ i 2 πl ( r ) ] exp [ i 2 πl ( r ) ] + exp [ i 2 πl ( r ) ] exp [ i 2 πl ( r ) ] + exp [ i 2 πl ( r ) ] exp [ i 2 πl ( r ) ] .
exp [ i 2 πl ( r ) ] exp [ i 2 πl ( r ) ]
= exp { i 2 π [ ( δ ( r r ) δ ( r r ) ) l ( r ) d r ] }
= Ψ l [ δ ( r r ) δ ( r r ) ] .
Ψ l [ δ ( r r ) δ ( r r ) ] = exp { 2 π 2 [ δ ( r r ) δ ( r r ) ] K l ( r , r ) [ δ ( r r ) δ ( r r ) ] d r d r } = exp { 2 π 2 [ K l ( r , r ) + K l ( r , r ) K l ( r , r ) K l ( r , r ) ] } .
R RR o ( r , r ) = 1 2 exp { 2 π 2 [ K l ( r , r ) + K l ( r , r ) K l ( r , r ) K l ( r , r ) ] } + exp { 2 π 2 [ K l ( r , r ) + K l ( r , r ) + K l ( r , r ) + K l ( r , r ) ] .
f R ( r ) = exp [ 2 π 2 K l ( 0 ) ] = m R
R RR o ( Δ r o ) = m R 2 2 [ R α ( Δ r o ) + R β ( Δ r o ) ]
h coh, R ( x i x o ) = T R ( ξ ) exp [ i 2 π ( x i x o ) ξ ] d ξ ,
h coh,I ( x i x o ) = T I ( ξ ) exp [ i 2 π ( x i x o ) ξ ] d ξ ,
T R ( ξ ) = T ( m T ξ ) exp [ i 2 π W o ( ξ ) ] cos [ 2 π W e ( ξ ) ]
T I ( ξ ) = T ( m T ξ ) exp [ i 2 π W o ( ξ ) ] sin [ 2 π W e ( ξ ) ]
c R = g R ( x i ) = 1 m T h coh , R ( x i x o ) f R ( x o m T ) d x o
= m R m T T R ( ξ ) exp [ i 2 π ( x i x o ) ξ ] d ξ d x o
= m R m T T R ( 0 )
= m R m T cos [ 2 π W e ( 0 ) ] .
R RR i ( x i , x i )
= 1 m T 2 ∫∫ h coh , R ( x i x o 1 ) h coh , R ( x i x o 2 ) f R ( x o 1 m T ) f R ( x o 2 m T ) d x o 1 d x o 2
+ 1 m T 2 ∫∫ h coh , I ( x i x o 1 ) h coh , I ( x i x o 2 ) f I ( x o 1 m T ) f I ( x o 2 m T ) d x o 1 d x o 2 ,
R II i ( x i , x i )
= 1 m T 2 h coh , R ( x i x o 1 ) h coh , R ( x i x o 2 ) f I ( x o 1 m T ) f I ( x o 2 m T ) d x o 1 d x o 2
+ 1 m T 2 h coh , I ( x i x o 1 ) h coh , I ( x i x o 2 ) f R ( x o 1 m T ) f R ( x o 2 m T ) d x o 1 d x o 2 ,
R RI i ( x i , x i )
= 1 m T 2 ∫∫ h coh , R ( x i x o 1 ) h coh , I ( x i x o 2 ) f R ( x o 1 m T ) f R ( x o 2 m T ) d x o 1 d x o 2
1 m T 2 ∫∫ h coh , I ( x i x o 1 ) h coh , R ( x i x o 2 ) f I ( x o 1 m T ) f I ( x o 2 m T ) d x o 1 d x o 2 ,
1 m T 2 ∫∫ h coh , R ( x i x o 1 ) h R ( x i x o 2 ) f R ( x o 1 m T ) f R ( x o 2 m T ) d x o 1 d x o 2
= 1 m T 2 T R ( ξ 1 ) exp [ i 2 π ( x i x o 1 ) ξ 1 ] d ξ 1
T R ( ξ 2 ) exp [ 2 π j ( x i x o 2 ) ξ 2 ] d ξ 2 f R ( x o 1 m T ) f R ( x o 2 m T ) d x o 1 d x o 2 .
= 1 m T 2 T R ( ξ 1 ) exp [ i 2 π ( x i x o 1 ) ξ 1 ] d ξ 1
T R ( ξ 2 ) exp [ i 2 π ( x i x o 1 Δ x o ) ξ 2 ] d ξ 2 R RR o ( Δ x o m T ) d x o 1 d Δ x o
= 1 m T 2 T R ( ξ 1 ) T R ( ξ 2 ) exp [ i 2 π ( x i ξ 1 + x i ξ 2 ) ]
× exp [ i 2 π ( ξ 2 + ξ 1 ) x o 1 ] d x 01 exp [ i 2 π Δ x o ξ 2 ] R RR o ( Δ x o m T ) d Δ x o d ξ 1 d ξ 2
= F Δ x [ T R ( ξ 1 ) F ξ 1 1 [ R RR o ( Δ x o m T ) ] ] .
R RR i ( Δ x i ) = 1 m T 2 F Δ x i { T R ( ξ ) T R ( ξ ) F ξ 1 [ R RR o ( Δ x o m T ) ] } + F Δ x i { T I ( ξ ) T I ( ξ 1 ) F ξ 1 [ R II o ( Δ x o m T ) ] } ,
R II i ( Δ x i ) = 1 m T 2 F Δ x i { T I ( ξ ) T I ( ξ ) F ξ 1 [ R RR o ( Δ x o m T ) ] } + F Δ x i { T R ( ξ ) T R ( ξ ) F ξ 1 [ R II o ( Δ x o m T ) ] } ,
R RI i ( Δ x i ) = 1 m T 2 F Δ x i { T R ( ξ ) T I ( ξ ) F ξ 1 [ R RR o ( Δ x o m T ) ] } F Δ x i { T I ( ξ ) T R ( ξ ) F ξ 1 [ R II o ( Δ x o m T ) ] } ,
R RR i ( Δ x i )
= m R 2 2 m T 2 F Δ x i { [ T R ( ξ ) T R ( ξ ) + T I ( ξ ) T I ( ξ ) ] F ξ 1 [ R α ( Δ x o m T ) ] } + F Δ x i { [ T R ( ξ ) T R ( ξ ) T I ( ξ ) T I ( ξ ) ] F ξ 1 [ R β ( Δ x o m T ) ] } ,
R II i ( Δ x i )
= m R 2 2 m T 2 F Δ x i { [ T R ( ξ ) T R ( ξ ) + T I ( ξ ) T I ( ξ ) ] F ξ 1 [ R α ( Δ x o m T ) ] } F Δ x i { [ T R ( ξ ) T R ( ξ ) T I ( ξ ) T I ( ξ ) ] F ξ 1 [ R β ( Δ x o m T ) ] } ,
R RI i ( Δ x i )
= m R 2 2 m T 2 F Δ x i { [ T R ( ξ ) T I ( ξ ) T I ( ξ ) T R ( ξ ) ] F ξ 1 [ R α ( Δ x o m T ) ] + F Δ x i { [ T R ( ξ ) T I ( ξ ) + T I ( ξ ) T R ( ξ ) ] F ξ 1 [ R β ( Δ x o m T ) ] ,
R IR i ( Δ x i )
= m R 2 2 m T 2 F Δ x i { [ T I ( ξ ) T R ( ξ ) T R ( ξ ) T I ( ξ ) ] F ξ 1 [ R α ( Δ x o m T ) ] } + F Δ x i { [ T I ( ξ ) T R ( ξ ) + T R ( ξ ) T I ( ξ ) ] F ξ 1 [ R β ( Δ x o m T ) ] }
K s ab ( 0 ) W e ( ξ ) = { sin [ 4 π W e ( ξ ) ] J ( ξ ) d ξ } cos [ π W e ( ξ ) ] J ( ξ )
{ sin [ 4 π W e ( ξ ) ] J ( ξ ) d ξ } sin [ π W e ( ξ ) ] J ( ξ ) ,
Q ( ξ , ξ ) = 2 K s ab ( 0 ) W e ( ξ ) W e ( ξ )
= sin [ 4 π W e ( ξ ) ] J ( ξ ) · sin [ 4 π W e ( ξ ) ] J ( ξ )
{ cos [ 4 π W e ( ξ ) ] J ( ξ ) d ξ } cos [ π W e ( ξ ) ] J ( ξ ) δ ( ξ ξ )
+ cos [ 4 π W e ( ξ ) ] J ( ξ ) · cos [ 4 π W e ( ξ ) ] J ( ξ )
{ sin [ 4 π W e ( ξ ) ] J ( ξ ) d ξ } sin [ 4 π W e ( ξ ) ] J ( ξ ) δ ( ξ ξ )
( p , Qp ) = ( cos [ 4 π W e ] J , p ) 2 ( cos [ 4 π W e ] , J , u ) cos [ 4 π W e ] , J , p 2 )
+ sin [ 4 π W e ] J , p ) 2 ( sin [ 4 π W e ] , J , u ) sin [ 4 π W e ] , J , p 2 )
J p 2 J u ( J , p 2 ) ,
q ( x ) r ( x ) d x 2 q ( x ) 2 d x r ( x ) 2 d x
J p { J p J u } .

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