Abstract

We suggest that, in the paraxial region, a double-plane symmetric optical system (anamorphic system) can be treated as two associated rotationally symmetric optical systems (RSOS). We find that paraxial quantities in the anamorphic system can be expressed as linear combinations of the paraxial marginal and chief rays traced in the two associated RSOS. As a result, we provide a set of equations that are key to derive the primary aberration coefficients for various anamorphic optical system types. By applying the generalized Aldis theorem to anamorphic optical systems, we build up the anamorphic total ray aberration equations. These equations can be reduced to third-order form, that is, the anamorphic primary ray aberration equations. We find that the terms in the anamorphic primary ray aberration equa tions can be expressed as paraxial marginal and chief ray-trace data in the two associated RSOS, together with normalized object and stop coordinates. More importantly, we build up a novel method for deriving the anamorphic primary aberration coefficients for anamorphic optical systems of various types.

© 2009 Optical Society of America

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References

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  1. H. A. Buchdahl, An Introduction to Hamiltonian Optics (Cambridge, 1970).
  2. J. M. Sasian, “How to approach the design of a bilateral symmetric optical system,” Opt. Eng. 33, 2045-2061 (1994).
    [CrossRef]
  3. W. T. Welford, Aberrations of Optical System (Adam Hilger, 1986).
  4. T. Kasuya, T. Suzuki, and K. Shimoda, “A prism anamorphic system for Gaussian beam expander,” Appl. Phys. 17, 131-136(1978).
    [CrossRef]
  5. J.-H. Jung and J.-W. Lee, “Anamorphic lens for a CCD camera apparatus,” U.S. patent 5,671,093 (23 September 1997).
  6. I. A. Neil, “Anamorphic imaging system,” U.S. patent 7,085,066 (1 August 2006).
  7. J. Gauvin, M. Doucet, M. Wang, S. Thibault, and B. Blanc, “Development of new family of wide-angle anamorphic lens with controlled distortion profile,” Proc. SPIE 5874, 1-12 (2005).
  8. E. Abbe, as cited by von Rohr, The Formation of Images in Optical Instruments (H. M. Stationary Office, 1920).
  9. G. J. Burch, “Some uses of cylindrical lens-systems, including rotation of images,” Proc. R. Soc. London 73, 281-286 (1904).
    [CrossRef]
  10. H. Chretien, “Anamorphotic lens system and method of making the same,” U.S. patent 1,962,892 (12 June 1934).
  11. R. K. Luneburg, Mathematical Theory of Optics, Chap. 4(University of California, 1966).
  12. P. J. Sands, “First-order optics of the general optical system,” J. Opt. Soc. Am. 62, 369 (1972).
    [CrossRef]
  13. B. D. Stone and G. W. Forbes, “Characterization of first-order optical properties for asymmetric systems,” J. Opt. Soc. Am. A 9, 478 (1992).
    [CrossRef]
  14. M. Herzberger, “First-order laws in asymmetrical optical systems. I. The image of a given congruence: fundamental conceptions,” J. Opt. Soc. Am. 26, 354 (1936).
    [CrossRef]
  15. M. Herzberger, “First-order laws in asymmetrical optical systems. II. The image congruences belonging to the rays emerging from a point in object and image space: fundamental forms,” J. Opt. Soc. Am. 26, 389 (1936).
    [CrossRef]
  16. H. A. Buchdahl, “Systems without symmetries: foundations of a theory of Langrangian aberration coefficients,” J. Opt. Soc. Am. 62, 1314 (1972).
    [CrossRef]
  17. J. Barcala, M. C. Vazquez, and A. Garcia, “Optic systems with spherical, cylindrical, and toric surfaces,” Appl. Opt. 34, 4900 (1995).
    [CrossRef] [PubMed]
  18. H. H. Arsenault, “A matrix representation for non-symmetrical optical systems,” J. Opt. 11, 87 (1980).
  19. M. C. Vacquez and J. Barcala, “Image formation and processing with toric surfaces. I. Geometrical optic properties,” J. Mod. Opt. 31, 947-958 (1984).
  20. I. Moreno, C. Ferreira, and M. M. Sánchez-López, “Ray matrix analysis of anamorphic fractional Fourier systems,” J. Opt. A Pure Appl. Opt. 8, 427 (2006).
    [CrossRef]
  21. C. G. Wynne, “The primary aberrations of anamorphic lens systems,” Proc. Phys. Soc. B 67, 529-537 (1954).
    [CrossRef]
  22. K. Bruder, “Die Bildfehler Dritter Ordnung in anamorphotischen Systemen,” Optik (Jena) 17, 663-670 (1960).
  23. G. G. Slyusarev, Aberration and Optical Design Theory (Adam Hilger, 1984).
  24. C. Chen and L. He, “The calculation of primary aberration of a torus,” Optik 87, 115-117 (1991).
  25. L. He and C. Chen, “The primary aberration coefficients of a torus,” Optik 94, 167-172 (1993).
  26. V. N. Mahajan, Optical Imaging and Aberrations. Part I. Ray Geometrical Optics (SPIE, 1998).
  27. A. Cox, A System of Optical Design (GB: The Focal Press, 1964).
  28. J. M. Sasian, “Double-curvature surfaces in mirror system design,” Opt. Eng. 36, 183-188 (1997).
    [CrossRef]
  29. R. R. Shannon, The Art and Science of Optical Design (Cambridge University, 1997).
  30. S. Yuan, “Aberrations of anamorphic optical systems,” Ph.D. dissertation (University of Arizona, 2008).
  31. J. C. Burfoot, “Third-order aberrations of doubly symmetric systems,” Proc. Phys. Soc. B 67, 523-528 (1954).
    [CrossRef]
  32. R. Barakat and A. Houston, “The aberrations on non-rotationally symmetric systems and their diffraction effects,” Opt. Acta 13, 1-30 (1966).
    [CrossRef]
  33. H. H. Hopkins, Wave Theory of Aberrations (Oxford University, 1950).

2006 (1)

I. Moreno, C. Ferreira, and M. M. Sánchez-López, “Ray matrix analysis of anamorphic fractional Fourier systems,” J. Opt. A Pure Appl. Opt. 8, 427 (2006).
[CrossRef]

2005 (1)

J. Gauvin, M. Doucet, M. Wang, S. Thibault, and B. Blanc, “Development of new family of wide-angle anamorphic lens with controlled distortion profile,” Proc. SPIE 5874, 1-12 (2005).

1997 (1)

J. M. Sasian, “Double-curvature surfaces in mirror system design,” Opt. Eng. 36, 183-188 (1997).
[CrossRef]

1995 (1)

1994 (1)

J. M. Sasian, “How to approach the design of a bilateral symmetric optical system,” Opt. Eng. 33, 2045-2061 (1994).
[CrossRef]

1993 (1)

L. He and C. Chen, “The primary aberration coefficients of a torus,” Optik 94, 167-172 (1993).

1992 (1)

1991 (1)

C. Chen and L. He, “The calculation of primary aberration of a torus,” Optik 87, 115-117 (1991).

1984 (1)

M. C. Vacquez and J. Barcala, “Image formation and processing with toric surfaces. I. Geometrical optic properties,” J. Mod. Opt. 31, 947-958 (1984).

1980 (1)

H. H. Arsenault, “A matrix representation for non-symmetrical optical systems,” J. Opt. 11, 87 (1980).

1978 (1)

T. Kasuya, T. Suzuki, and K. Shimoda, “A prism anamorphic system for Gaussian beam expander,” Appl. Phys. 17, 131-136(1978).
[CrossRef]

1972 (2)

1966 (1)

R. Barakat and A. Houston, “The aberrations on non-rotationally symmetric systems and their diffraction effects,” Opt. Acta 13, 1-30 (1966).
[CrossRef]

1960 (1)

K. Bruder, “Die Bildfehler Dritter Ordnung in anamorphotischen Systemen,” Optik (Jena) 17, 663-670 (1960).

1954 (2)

C. G. Wynne, “The primary aberrations of anamorphic lens systems,” Proc. Phys. Soc. B 67, 529-537 (1954).
[CrossRef]

J. C. Burfoot, “Third-order aberrations of doubly symmetric systems,” Proc. Phys. Soc. B 67, 523-528 (1954).
[CrossRef]

1936 (2)

1904 (1)

G. J. Burch, “Some uses of cylindrical lens-systems, including rotation of images,” Proc. R. Soc. London 73, 281-286 (1904).
[CrossRef]

Abbe, E.

E. Abbe, as cited by von Rohr, The Formation of Images in Optical Instruments (H. M. Stationary Office, 1920).

Arsenault, H. H.

H. H. Arsenault, “A matrix representation for non-symmetrical optical systems,” J. Opt. 11, 87 (1980).

Barakat, R.

R. Barakat and A. Houston, “The aberrations on non-rotationally symmetric systems and their diffraction effects,” Opt. Acta 13, 1-30 (1966).
[CrossRef]

Barcala, J.

J. Barcala, M. C. Vazquez, and A. Garcia, “Optic systems with spherical, cylindrical, and toric surfaces,” Appl. Opt. 34, 4900 (1995).
[CrossRef] [PubMed]

M. C. Vacquez and J. Barcala, “Image formation and processing with toric surfaces. I. Geometrical optic properties,” J. Mod. Opt. 31, 947-958 (1984).

Blanc, B.

J. Gauvin, M. Doucet, M. Wang, S. Thibault, and B. Blanc, “Development of new family of wide-angle anamorphic lens with controlled distortion profile,” Proc. SPIE 5874, 1-12 (2005).

Bruder, K.

K. Bruder, “Die Bildfehler Dritter Ordnung in anamorphotischen Systemen,” Optik (Jena) 17, 663-670 (1960).

Buchdahl, H. A.

Burch, G. J.

G. J. Burch, “Some uses of cylindrical lens-systems, including rotation of images,” Proc. R. Soc. London 73, 281-286 (1904).
[CrossRef]

Burfoot, J. C.

J. C. Burfoot, “Third-order aberrations of doubly symmetric systems,” Proc. Phys. Soc. B 67, 523-528 (1954).
[CrossRef]

Chen, C.

L. He and C. Chen, “The primary aberration coefficients of a torus,” Optik 94, 167-172 (1993).

C. Chen and L. He, “The calculation of primary aberration of a torus,” Optik 87, 115-117 (1991).

Chretien, H.

H. Chretien, “Anamorphotic lens system and method of making the same,” U.S. patent 1,962,892 (12 June 1934).

Cox, A.

A. Cox, A System of Optical Design (GB: The Focal Press, 1964).

Doucet, M.

J. Gauvin, M. Doucet, M. Wang, S. Thibault, and B. Blanc, “Development of new family of wide-angle anamorphic lens with controlled distortion profile,” Proc. SPIE 5874, 1-12 (2005).

Ferreira, C.

I. Moreno, C. Ferreira, and M. M. Sánchez-López, “Ray matrix analysis of anamorphic fractional Fourier systems,” J. Opt. A Pure Appl. Opt. 8, 427 (2006).
[CrossRef]

Forbes, G. W.

Garcia, A.

Gauvin, J.

J. Gauvin, M. Doucet, M. Wang, S. Thibault, and B. Blanc, “Development of new family of wide-angle anamorphic lens with controlled distortion profile,” Proc. SPIE 5874, 1-12 (2005).

He, L.

L. He and C. Chen, “The primary aberration coefficients of a torus,” Optik 94, 167-172 (1993).

C. Chen and L. He, “The calculation of primary aberration of a torus,” Optik 87, 115-117 (1991).

Herzberger, M.

Hopkins, H. H.

H. H. Hopkins, Wave Theory of Aberrations (Oxford University, 1950).

Houston, A.

R. Barakat and A. Houston, “The aberrations on non-rotationally symmetric systems and their diffraction effects,” Opt. Acta 13, 1-30 (1966).
[CrossRef]

Jung, J.-H.

J.-H. Jung and J.-W. Lee, “Anamorphic lens for a CCD camera apparatus,” U.S. patent 5,671,093 (23 September 1997).

Kasuya, T.

T. Kasuya, T. Suzuki, and K. Shimoda, “A prism anamorphic system for Gaussian beam expander,” Appl. Phys. 17, 131-136(1978).
[CrossRef]

Lee, J.-W.

J.-H. Jung and J.-W. Lee, “Anamorphic lens for a CCD camera apparatus,” U.S. patent 5,671,093 (23 September 1997).

Luneburg, R. K.

R. K. Luneburg, Mathematical Theory of Optics, Chap. 4(University of California, 1966).

Mahajan, V. N.

V. N. Mahajan, Optical Imaging and Aberrations. Part I. Ray Geometrical Optics (SPIE, 1998).

Moreno, I.

I. Moreno, C. Ferreira, and M. M. Sánchez-López, “Ray matrix analysis of anamorphic fractional Fourier systems,” J. Opt. A Pure Appl. Opt. 8, 427 (2006).
[CrossRef]

Neil, I. A.

I. A. Neil, “Anamorphic imaging system,” U.S. patent 7,085,066 (1 August 2006).

Sánchez-López, M. M.

I. Moreno, C. Ferreira, and M. M. Sánchez-López, “Ray matrix analysis of anamorphic fractional Fourier systems,” J. Opt. A Pure Appl. Opt. 8, 427 (2006).
[CrossRef]

Sands, P. J.

Sasian, J. M.

J. M. Sasian, “Double-curvature surfaces in mirror system design,” Opt. Eng. 36, 183-188 (1997).
[CrossRef]

J. M. Sasian, “How to approach the design of a bilateral symmetric optical system,” Opt. Eng. 33, 2045-2061 (1994).
[CrossRef]

Shannon, R. R.

R. R. Shannon, The Art and Science of Optical Design (Cambridge University, 1997).

Shimoda, K.

T. Kasuya, T. Suzuki, and K. Shimoda, “A prism anamorphic system for Gaussian beam expander,” Appl. Phys. 17, 131-136(1978).
[CrossRef]

Slyusarev, G. G.

G. G. Slyusarev, Aberration and Optical Design Theory (Adam Hilger, 1984).

Stone, B. D.

Suzuki, T.

T. Kasuya, T. Suzuki, and K. Shimoda, “A prism anamorphic system for Gaussian beam expander,” Appl. Phys. 17, 131-136(1978).
[CrossRef]

Thibault, S.

J. Gauvin, M. Doucet, M. Wang, S. Thibault, and B. Blanc, “Development of new family of wide-angle anamorphic lens with controlled distortion profile,” Proc. SPIE 5874, 1-12 (2005).

Vacquez, M. C.

M. C. Vacquez and J. Barcala, “Image formation and processing with toric surfaces. I. Geometrical optic properties,” J. Mod. Opt. 31, 947-958 (1984).

Vazquez, M. C.

Wang, M.

J. Gauvin, M. Doucet, M. Wang, S. Thibault, and B. Blanc, “Development of new family of wide-angle anamorphic lens with controlled distortion profile,” Proc. SPIE 5874, 1-12 (2005).

Welford, W. T.

W. T. Welford, Aberrations of Optical System (Adam Hilger, 1986).

Wynne, C. G.

C. G. Wynne, “The primary aberrations of anamorphic lens systems,” Proc. Phys. Soc. B 67, 529-537 (1954).
[CrossRef]

Yuan, S.

S. Yuan, “Aberrations of anamorphic optical systems,” Ph.D. dissertation (University of Arizona, 2008).

Appl. Opt. (1)

Appl. Phys. (1)

T. Kasuya, T. Suzuki, and K. Shimoda, “A prism anamorphic system for Gaussian beam expander,” Appl. Phys. 17, 131-136(1978).
[CrossRef]

J. Mod. Opt. (1)

M. C. Vacquez and J. Barcala, “Image formation and processing with toric surfaces. I. Geometrical optic properties,” J. Mod. Opt. 31, 947-958 (1984).

J. Opt. (1)

H. H. Arsenault, “A matrix representation for non-symmetrical optical systems,” J. Opt. 11, 87 (1980).

J. Opt. A Pure Appl. Opt. (1)

I. Moreno, C. Ferreira, and M. M. Sánchez-López, “Ray matrix analysis of anamorphic fractional Fourier systems,” J. Opt. A Pure Appl. Opt. 8, 427 (2006).
[CrossRef]

J. Opt. Soc. Am. (4)

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

Opt. Acta (1)

R. Barakat and A. Houston, “The aberrations on non-rotationally symmetric systems and their diffraction effects,” Opt. Acta 13, 1-30 (1966).
[CrossRef]

Opt. Eng. (2)

J. M. Sasian, “Double-curvature surfaces in mirror system design,” Opt. Eng. 36, 183-188 (1997).
[CrossRef]

J. M. Sasian, “How to approach the design of a bilateral symmetric optical system,” Opt. Eng. 33, 2045-2061 (1994).
[CrossRef]

Optik (2)

C. Chen and L. He, “The calculation of primary aberration of a torus,” Optik 87, 115-117 (1991).

L. He and C. Chen, “The primary aberration coefficients of a torus,” Optik 94, 167-172 (1993).

Optik (Jena) (1)

K. Bruder, “Die Bildfehler Dritter Ordnung in anamorphotischen Systemen,” Optik (Jena) 17, 663-670 (1960).

Proc. Phys. Soc. B (2)

J. C. Burfoot, “Third-order aberrations of doubly symmetric systems,” Proc. Phys. Soc. B 67, 523-528 (1954).
[CrossRef]

C. G. Wynne, “The primary aberrations of anamorphic lens systems,” Proc. Phys. Soc. B 67, 529-537 (1954).
[CrossRef]

Proc. R. Soc. London (1)

G. J. Burch, “Some uses of cylindrical lens-systems, including rotation of images,” Proc. R. Soc. London 73, 281-286 (1904).
[CrossRef]

Proc. SPIE (1)

J. Gauvin, M. Doucet, M. Wang, S. Thibault, and B. Blanc, “Development of new family of wide-angle anamorphic lens with controlled distortion profile,” Proc. SPIE 5874, 1-12 (2005).

Other (13)

E. Abbe, as cited by von Rohr, The Formation of Images in Optical Instruments (H. M. Stationary Office, 1920).

H. Chretien, “Anamorphotic lens system and method of making the same,” U.S. patent 1,962,892 (12 June 1934).

R. K. Luneburg, Mathematical Theory of Optics, Chap. 4(University of California, 1966).

W. T. Welford, Aberrations of Optical System (Adam Hilger, 1986).

J.-H. Jung and J.-W. Lee, “Anamorphic lens for a CCD camera apparatus,” U.S. patent 5,671,093 (23 September 1997).

I. A. Neil, “Anamorphic imaging system,” U.S. patent 7,085,066 (1 August 2006).

H. A. Buchdahl, An Introduction to Hamiltonian Optics (Cambridge, 1970).

H. H. Hopkins, Wave Theory of Aberrations (Oxford University, 1950).

G. G. Slyusarev, Aberration and Optical Design Theory (Adam Hilger, 1984).

R. R. Shannon, The Art and Science of Optical Design (Cambridge University, 1997).

S. Yuan, “Aberrations of anamorphic optical systems,” Ph.D. dissertation (University of Arizona, 2008).

V. N. Mahajan, Optical Imaging and Aberrations. Part I. Ray Geometrical Optics (SPIE, 1998).

A. Cox, A System of Optical Design (GB: The Focal Press, 1964).

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Figures (4)

Fig. 1
Fig. 1

Anamorphic system made from cross-cylindrical lenses.

Fig. 2
Fig. 2

Direction cosines.

Fig. 3
Fig. 3

Three-dimensional ray tracing.

Fig. 4
Fig. 4

Ray tracing on surface j.

Equations (55)

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

{ L = cos ( A O P ) = O A O P = C P O P M = cos ( B O P ) = O B O P = C P O P N = cos ( C O P ) = O C O P .
L 2 + M 2 + N 2 = 1.
N = [ 1 ( L 2 + M 2 ) ] 1 / 2 .
N = 1 ( L 2 + M 2 ) 2 + .
{ L = cos ( A O P ) = C P O P = C P O C = tan ( C O P ) = u x M = cos ( B O P ) = C P O P = C P O C = tan ( C O P ) = u y N = 1 ,
z = 1 2 ( x 2 r x + y 2 r y ) ,
F ( x , y , z ) = z 1 2 ( x 2 r x + y 2 r y ) = 0.
( α , β , γ ) = ( F x , F y , F z ) { ( F x ) 2 + ( F y ) 2 + ( F z ) 2 } 1 2 ,
{ α = c x x β = c y y γ = 1 ,
x j x j 1 L j 1 = y j y j 1 M j 1 = z j z j 1 + t j 1 N j 1 ,
n j L j n j 1 L j 1 α j = n j M j n j 1 M j 1 β j = n j N j n j 1 N j 1 γ j ,
{ x j x j 1 u x , j 1 = y j y j 1 u y , j 1 = t j 1 n j u x , j n j 1 u x , j 1 c x , j x j = n j u y , j n j 1 u y , j 1 c y , j y j = ( n j n j 1 ) .
{ x j x j 1 = t j 1 u x , j 1 n j u x , j n j 1 u x , j 1 = x j ( n j n j 1 ) c x , j .
{ y j y j 1 = t j 1 u y , j 1 n j u y , j n j 1 u y , j 1 = y j ( n j n j 1 ) c y , j .
{ h ¯ ¯ = C h + D h ¯ u ¯ ¯ = C u + D u ¯ ,
{ x ¯ ¯ j = ρ x h x , j + H x h ¯ x , j u ¯ ¯ x , j = ρ x u x , j + H x u ¯ x , j y ¯ ¯ j = ρ y h y , j + H y h ¯ y , j u ¯ ¯ y , j = ρ y u y , j + H y u ¯ y , j ,
n j 1 u x , j 1 + n j 1 h x , j c x , j = n j u x , j + n j h x , j c x , j = A x , j .
n j 1 u ¯ x , j 1 + n j 1 h ¯ x , j c x , j = n j u ¯ x , j + n j h ¯ x , j c x , j = A ¯ x , j .
{ n j 1 u y , j 1 + n j 1 h y , j c y , j = n j u u , j + n j h y , j c y , j = A y , j n j 1 u ¯ y , j 1 + n j 1 h ¯ y , j c y , j = n j u ¯ y , j + n j h ¯ y , j c y , j = A ¯ y , j .
{ ψ x = n j ( h ¯ x , j u x , j h x , j u ¯ x , j ) = A x , j h ¯ x , j A ¯ x , j h x , j ψ y = n j ( h ¯ y , j u y , j h y , j u ¯ y , j ) = A y , j h ¯ y , j A ¯ y , j h y , j .
W ( 4 ) = D 1 ρ x 4 + D 2 ρ y 4 + D 3 ρ x 2 ρ y 2 + D 4 H x ρ x 3 + D 5 H y ρ x 2 ρ y + D 6 H x ρ x ρ y 2 + D 7 H y ρ y 3 + D 8 H x 2 ρ x 2 + D 9 H y 2 ρ y 2 + D 10 H y 2 ρ x 2 + D 11 H x 2 ρ y 2 + D 12 H x H y ρ x ρ y + D 13 H x 3 ρ x + D 14 H y 3 ρ y + D 15 H x H y 2 ρ x + D 16 H x 2 H y ρ y ,
{ δ ξ k 3 = 1 n k u x , k W ρ x δ η k 3 = 1 n k u y , k W ρ y ,
{ δ ξ k 3 = ( 4 D 1 ρ x 3 + 2 D 3 ρ x ρ y 2 + 3 D 4 H x ρ x 2 + 2 D 5 H y ρ x ρ y + D 6 H x ρ y 2 + 2 D 8 H x 2 ρ x + 2 D 10 H y 2 ρ x + D 12 H x H y ρ y + D 13 H x 3 + D 15 H x H y 2 ) / n k u x , k δ η k 3 = ( 4 D 2 ρ y 3 + 2 D 3 ρ x 2 ρ y + D 5 H y ρ x 2 + 2 D 6 H x ρ x ρ y + 3 D 7 H y ρ y 2 + 2 D 9 H y 2 ρ y + 2 D 11 H x 2 ρ y + D 12 H x H y ρ x + D 14 H y 3 + D 16 H x 2 H y ) / n k u y , k .
ξ = x + L N ( l x z ) .
η = y + M N ( l y z ) ,
n u x N ξ = n u x x N + n u x l x L n u x z L .
n u x N ξ = ( A x n h x c x ) x N ( A x n h x c x ) z L + n u x l x L .
n u x N ξ 0 = N H x ψ x ,
n u x N δ ξ = A x ( x N z L ) n h x c x ( x N z L ) n h x L ψ x H x N = ( A x x H x ψ x ) N A x z L h x c x x n N + ( h x c x z h x ) n L ,
Δ { n u x N δ ξ } = ( A x x H x ψ x ) Δ N A x z Δ L h x c x x Δ n N + ( h x c x z h x ) Δ n L .
Δ { n u y N δ η } = ( A y y H y ψ y ) Δ N A y z Δ M h y c y y Δ n N + ( h y c y z h y ) Δ n M .
{ Δ n L = α γ Δ n N Δ n M = β γ Δ n N .
Δ { n u x N δ ξ } = ( A x x H x ψ x ) Δ N A x z Δ L + [ ( h x c x z h x ) α γ h x c x x ] Δ n N .
δ ξ k = j = 1 k A x , j ( x j Δ N j z j Δ L j ) j = 1 k H x ψ x Δ N j + j = 1 k [ ( h x , j c x , j z j h x , j ) α j γ j h x , j c x , j x j ] Δ n j N j / n k u x , k N k .
δ η k = j = 1 k A y , j ( y j Δ N j z j Δ M j ) j = 1 k H y ψ y Δ N j + j = 1 k [ ( h y , j c y , j z j h y , j ) β j γ j h y , j c y , j y j ] Δ n j N j / n k u y , k N k .
z j = 1 2 ( x j 2 r x , j + y j 2 r y , j ) + 1 8 ( x j 4 r 3 , j 3 + 2 x j 2 y j 2 r 4 , j 3 + y j 4 r 5 , j 3 ) ,
{ α j γ j = ( x j r x , j + x j 3 2 r 3 , j 3 + x j y j 2 2 r 4 , j 3 ) β j γ j = ( y j r y , j + y j 3 2 r 5 , j 3 + x j 2 y j 2 r 4 , j 3 ) .
{ x j = x j 1 = x ¯ ¯ j y j = y j 1 = y ¯ ¯ j L j = L j 1 = u ¯ ¯ x , j M j = M j 1 = u ¯ ¯ y , j ,
N j = 1 L j 2 + M j 2 2 + = 1 δ N j ,
δ N j = L j 2 + M j 2 2 + .
δ N j = δ N j 2 = L j 1 2 + M j 1 2 2 .
N j = N j 2 = 1 δ N j 2 ,
n j N j = n j N j 2 = n j n j δ N j 2 .
1 N k = 1 1 δ N k 2 = 1 + δ N k 2 .
z j = z j 2 = 1 2 ( x j 1 2 r x , j + y j 1 2 r y , j ) .
α j γ j = ( x j 1 r x , j + x j 1 3 2 r 3 , j 3 + x j 1 y j 1 2 2 r 4 , j 3 ) = α j 1 γ j 1 + α j 3 γ j 3 ,
{ α j 1 γ j 1 = ( x j 1 r x , j ) α j 3 γ j 3 = ( x j 1 3 2 r 3 , j 3 + x j 1 y j 1 2 2 r 4 , j 3 ) ,
β j γ j = ( y j 1 r y , j + y j 1 3 2 r 5 , j 3 + x j 1 2 y j 1 2 r 4 , j 3 ) = β j 1 γ j 1 + β j 3 γ j 3 ,
{ β j 1 γ j 1 = ( y j 1 r y , j ) β j 3 γ j 3 = ( y j 1 3 2 r 5 , j 3 + x j 1 2 y j 1 2 r 4 , j 3 ) .
{ δ ξ k 3 = Δ [ j = 1 k A x , j ( x j 1 δ N j 2 + z j 2 L j 1 ) + H x ψ x j = 1 k δ N j 2 j = 1 k h x , j ( c x , j 2 z j 2 x j 1 + α j 3 γ j 3 ) n j ] / n k u x , k δ η k 3 = Δ [ j = 1 k A y , j ( y j 1 δ N j 2 + z j 2 M j 1 ) + H y ψ y j = 1 k δ N j 2 j = 1 k h y , j ( c y , j 2 z j 2 y j 1 + β j 3 γ j 3 ) n j ] / n k u y , k .
{ x j 1 = x ¯ ¯ j = ρ x h x , j + H x h ¯ x , j y j 1 = y ¯ ¯ j = ρ y h y , j + H y h ¯ y , j L j 1 = u ¯ ¯ x , j = ρ x u x , j + H x u ¯ x , j M j 1 = u ¯ ¯ y , j = ρ y u y , j + H y u ¯ y , j ,
{ δ N j 2 = L j 1 2 + M j 1 2 2 = 1 2 [ ( ρ x u x , j + H x u ¯ x , j ) 2 + ( ρ y u y , j + H y u ¯ y , j ) 2 ] z j 2 = 1 2 ( x j 1 2 r x , j + y j 1 2 r y , j ) = 1 2 [ 1 r x , j ( ρ x h x , j + H x h ¯ x , j ) 2 + 1 r y , j ( ρ y h y , j + H y h ¯ y , j ) 2 ] α j 3 γ j 3 = ( x j 1 3 2 r 3 , j 3 + x j 1 y j 1 2 2 r 4 , j 3 ) = 1 2 [ ( ρ x h x , j + H x h ¯ x , j ) 3 r 3 , j 3 + ( ρ x h x , j + H x h ¯ x , j ) ( ρ y h y , j + H y h ¯ y , j ) 2 r 4 , j 3 ] β j 3 γ j 3 = ( y j 1 3 2 r 5 , j 3 + x j 1 2 y j 1 2 r 4 , j 3 ) = 1 2 [ ( ρ y h y , j + H y h ¯ y , j ) 3 r 5 , j 3 + ( ρ x h x , j + H x h ¯ x , j ) 2 ( ρ y h y , j + H y h ¯ y , j ) r 4 , j 3 ] .
{ x ¯ ¯ j = ρ x h x , j + H x h ¯ x , j u ¯ ¯ x , j = ρ x u x , j + H x u ¯ x , j y ¯ ¯ j = ρ y h y , j + H y h ¯ y , j u ¯ ¯ y , j = ρ y u y , j + H y u ¯ y , j .
{ δ ξ k 3 = Δ [ j = 1 k A x , j ( x j 1 δ N j 2 + z j 2 L j 1 ) + H x ψ x j = 1 k δ N j 2 j = 1 k h x , j ( c x , j 2 z j 2 x j 1 + α j 3 γ j 3 ) n j ] / n k u x , k δ η k 3 = Δ [ j = 1 k A y , j ( y j 1 δ N j 2 + z j 2 M j 1 ) + H y ψ y j = 1 k δ N j 2 j = 1 k h y , j ( c y , j 2 z j 2 y j 1 + β j 3 γ j 3 ) n j ] / n k u y , k .
{ δ ξ k 3 = ( 4 D 1 ρ x 3 + 2 D 3 ρ x ρ y 2 + 3 D 4 H x ρ x 2 + 2 D 5 H y ρ x ρ y + D 6 H x ρ y 2 + 2 D 8 H x 2 ρ x + 2 D 10 H y 2 ρ x + D 12 H x H y ρ y + D 13 H x 3 + D 15 H x H y 2 ) / n k u x , k δ η k 3 = ( 4 D 2 ρ y 3 + 2 D 3 ρ x 2 ρ y + D 5 H y ρ x 2 + 2 D 6 H x ρ x ρ y + 3 D 7 H y ρ y 2 + 2 D 9 H y 2 ρ y + 2 D 11 H x 2 ρ y + D 12 H x H y ρ x + D 14 H y 3 + D 16 H x 2 H y ) / n k u y , k .

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