Abstract

Methods are described here that can be used to determine expressions for the second-order properties (with respect to a particular ray—the base ray) of any system composed of individual homogeneous, isotropic regions. Since asymmetric systems cannot be expected to form the best image when the object and image planes are normal to the associated base ray segments, the significance of object and image tilts, to second order, is also considered. With these results, it is possible to determine constraints on a system’s configuration that ensure a given set of second-order imaging properties. As an illustrative example, constraints on the configuration of a single interface—either refracting or reflecting—are determined when sharp, second-order imagery is required. It is shown that the only nontrivial solutions are spherical refracting surfaces, and, for a given spherical surface, the basal object and image points must correspond to aplanatic points associated with the sphere.

© 1992 Optical Society of America

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References

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  1. B. D. Stone, G. W. Forbes, “Foundation of first-order layout for asymmetric systems: an application of Hamilton’s methods,” J. Opt. Soc. Am. A 9, 96–109 (1992).
    [CrossRef]
  2. B. D. Stone, G. W. Forbes, “First-order layout of asymmetric systems: sharp imagery of a single plane object,” J. Opt. Soc. Am. A 9, 832–843 (1992).
    [CrossRef]
  3. For a description of Hamilton’s characteristic functions see, for example, H. A. Buchdahl, Introduction to Hamiltonian Optics (Cambridge U. Press, Cambridge, 1970), Chap. 2.
  4. More precisely, these are restricted characteristics, but the distinction is essential only for the analysis presented in Appendix B. For a description of restricted characteristics see G. W. Forbes, “New class of characteristic functions in Hamiltonian optics,” J. Opt. Soc. Am. 72, 1698–1702 (1982). Note that, if the base surfaces were not planes or if the associated coordinate systems were not chosen with one of the axes normal to the associated base planes, then pand p′ appearing as arguments of the characteristic functions would not represent optical direction cosines.
    [CrossRef]
  5. This follows, for example, from Eqs. (2.10a) and (2.10b) of Ref. 1 or from Eqs. (2.3) here.
  6. In Ref. 1 it is established that there are no hidden constraints among these ten elements.
  7. G. W. Forbes, “Order doubling in the determination of characteristic functions,”J. Opt. Soc. Am. 72, 1097–1099 (1982).
    [CrossRef]
  8. G. W. Forbes, “Concatenation of restricted characteristic functions,”J. Opt. Soc. Am. 72, 1702–1706 (1982).
    [CrossRef]
  9. The value of the point-angle mixed characteristic, C01(y, p′), is the optical length along the ray whose point of intersection with the anterior base plane is specified by yand whose direction in image space is specified by p′. With the base planes and associated coordinate systems that are used here, the optical length is measured from the anterior base plane to a particular point along the base ray in image space that is chosen as follows: a line from this point to the coordinate origin in image space is perpendicular to the ray of interest. Since the origins of the original and rotated coordinate systems are coincident, a term analogous to expression (3.3) for image space is not required. For further descriptions of the physical interpretations of restricted characteristics, see the citation in Note 4.
  10. B. D. Stone, G. W. Forbes, “Characterization of first-order optical properties for asymmetric systems,” J. Opt. Soc. Am. A 9, 478–489 (1992).
    [CrossRef]
  11. If the term in parentheses in Eq. (3.4f) is denoted by ℙlmp, Eqs. (3.6a) and (3.6b) represent the requirements that (ℙ111− ℙ122) and (ℙ111+ ℙ122) vanish; Eqs. (3.6c) and (3.6d) represent the requirements that (ℙ211− ℙ222) and (ℙ211+ ℙ222) vanish; and Eqs. (3.6e) and (3.6f) represent the requirements that ℙ112and ℙ212) vanish.
  12. For a description of the Scheimpflug condition see, for example, R. Kingslake, Optical System Design (Academic, Orlando, Fla., 1983), Chap. 5, Sec. VI.
  13. B. D. Stone, G. W. Forbes, “Forms of the characteristic function for asymmetric systems that form sharp images to first order,” J. Opt. Soc. Am. A 9, 820–831 (1992).
    [CrossRef]
  14. There is a second solution:S22=-1t[1 cos(θs)-μ cos(θs′)].This solution is ruled out, however, since the resulting image is at infinity. That is, in this case, the object is at the front focal point. Also, as an aside, note that, when the base ray is incident normally upon the interface, it follows from Eqs. (4.10) that S12vanishes and S11and S22are equal. In this case, it also follows from Eqs. (4.13) and (4.14) that the elements ofSvanish. Consequently, as expected, when the base ray is incident normally upon the interface, the surface must be rotationally symmetric about the base ray in order for the system to possess sharp, second-order imagery.
  15. For a description of the aplanatic points of a sphere see, for example, R. K. Luneburg, Mathematical Theory of Optics(U. California Press, Los Angeles, Calif., 1964), Sec. 23.6. Briefly, for a given value of μ, there is a family of spherical surfaces that forms a perfect image of a given object point. The image lies upon the line containing the center of curvature of the interface and the object point. A given member of the family is distinguished from the other members by the distance, measured along the line containing the object point and the center of curvature of the interface, from the object point to the interface. If this distance is denoted by tc, the radius of curvature of the interface is given by tc/(μ+ 1).

1992 (4)

1982 (3)

Buchdahl, H. A.

For a description of Hamilton’s characteristic functions see, for example, H. A. Buchdahl, Introduction to Hamiltonian Optics (Cambridge U. Press, Cambridge, 1970), Chap. 2.

Forbes, G. W.

B. D. Stone, G. W. Forbes, “Foundation of first-order layout for asymmetric systems: an application of Hamilton’s methods,” J. Opt. Soc. Am. A 9, 96–109 (1992).
[CrossRef]

B. D. Stone, G. W. Forbes, “Characterization of first-order optical properties for asymmetric systems,” J. Opt. Soc. Am. A 9, 478–489 (1992).
[CrossRef]

B. D. Stone, G. W. Forbes, “Forms of the characteristic function for asymmetric systems that form sharp images to first order,” J. Opt. Soc. Am. A 9, 820–831 (1992).
[CrossRef]

B. D. Stone, G. W. Forbes, “First-order layout of asymmetric systems: sharp imagery of a single plane object,” J. Opt. Soc. Am. A 9, 832–843 (1992).
[CrossRef]

More precisely, these are restricted characteristics, but the distinction is essential only for the analysis presented in Appendix B. For a description of restricted characteristics see G. W. Forbes, “New class of characteristic functions in Hamiltonian optics,” J. Opt. Soc. Am. 72, 1698–1702 (1982). Note that, if the base surfaces were not planes or if the associated coordinate systems were not chosen with one of the axes normal to the associated base planes, then pand p′ appearing as arguments of the characteristic functions would not represent optical direction cosines.
[CrossRef]

G. W. Forbes, “Concatenation of restricted characteristic functions,”J. Opt. Soc. Am. 72, 1702–1706 (1982).
[CrossRef]

G. W. Forbes, “Order doubling in the determination of characteristic functions,”J. Opt. Soc. Am. 72, 1097–1099 (1982).
[CrossRef]

Kingslake, R.

For a description of the Scheimpflug condition see, for example, R. Kingslake, Optical System Design (Academic, Orlando, Fla., 1983), Chap. 5, Sec. VI.

Stone, B. D.

J. Opt. Soc. Am. (3)

J. Opt. Soc. Am. A (4)

Other (8)

For a description of Hamilton’s characteristic functions see, for example, H. A. Buchdahl, Introduction to Hamiltonian Optics (Cambridge U. Press, Cambridge, 1970), Chap. 2.

This follows, for example, from Eqs. (2.10a) and (2.10b) of Ref. 1 or from Eqs. (2.3) here.

In Ref. 1 it is established that there are no hidden constraints among these ten elements.

The value of the point-angle mixed characteristic, C01(y, p′), is the optical length along the ray whose point of intersection with the anterior base plane is specified by yand whose direction in image space is specified by p′. With the base planes and associated coordinate systems that are used here, the optical length is measured from the anterior base plane to a particular point along the base ray in image space that is chosen as follows: a line from this point to the coordinate origin in image space is perpendicular to the ray of interest. Since the origins of the original and rotated coordinate systems are coincident, a term analogous to expression (3.3) for image space is not required. For further descriptions of the physical interpretations of restricted characteristics, see the citation in Note 4.

If the term in parentheses in Eq. (3.4f) is denoted by ℙlmp, Eqs. (3.6a) and (3.6b) represent the requirements that (ℙ111− ℙ122) and (ℙ111+ ℙ122) vanish; Eqs. (3.6c) and (3.6d) represent the requirements that (ℙ211− ℙ222) and (ℙ211+ ℙ222) vanish; and Eqs. (3.6e) and (3.6f) represent the requirements that ℙ112and ℙ212) vanish.

For a description of the Scheimpflug condition see, for example, R. Kingslake, Optical System Design (Academic, Orlando, Fla., 1983), Chap. 5, Sec. VI.

There is a second solution:S22=-1t[1 cos(θs)-μ cos(θs′)].This solution is ruled out, however, since the resulting image is at infinity. That is, in this case, the object is at the front focal point. Also, as an aside, note that, when the base ray is incident normally upon the interface, it follows from Eqs. (4.10) that S12vanishes and S11and S22are equal. In this case, it also follows from Eqs. (4.13) and (4.14) that the elements ofSvanish. Consequently, as expected, when the base ray is incident normally upon the interface, the surface must be rotationally symmetric about the base ray in order for the system to possess sharp, second-order imagery.

For a description of the aplanatic points of a sphere see, for example, R. K. Luneburg, Mathematical Theory of Optics(U. California Press, Los Angeles, Calif., 1964), Sec. 23.6. Briefly, for a given value of μ, there is a family of spherical surfaces that forms a perfect image of a given object point. The image lies upon the line containing the center of curvature of the interface and the object point. A given member of the family is distinguished from the other members by the distance, measured along the line containing the object point and the center of curvature of the interface, from the object point to the interface. If this distance is denoted by tc, the radius of curvature of the interface is given by tc/(μ+ 1).

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Figures (4)

Fig. 1
Fig. 1

Schematic representation of a single-interface system. The base planes are taken here to be normal to the base ray.

Fig. 2
Fig. 2

Representation of a homogeneous region between two general surfaces. Coordinate systems associated with each surface are shown, as is the base ray.

Fig. 3
Fig. 3

Schematic illustration of a single, refracting surface that possesses sharp, second-order imagery. The interface is represented by a semicircle. The basal object point, the basal image point, and the center of curvature of the interface lie along the long, thin, dashed line. An extrapolation of the base ray segment in image space is illustrated as a thick, dashed line. The displacement, measured along the base ray, from the basal object point to the interface is labeled t. The displacement, measured along the line containing the base ray segment in image space, from the basal image point to the interface, is t/μ. The specific configuration chosen for this figure is θs = π/6 and μ = 1 / 2 (so that the absolute value of the magnification is 2).

Fig. 4
Fig. 4

The object shown in the top figure is imaged, to second order, as illustrated below it when θs = π/6 and μ = 1 / 2. The dashed lines in the image represent the image that would be formed in the absence of distortion.

Equations (145)

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C ρ σ ( u , u ) = c 0 + ( u i - u b , i ) f i ρ σ + ( u i - u b , i ) b i ρ σ + ½ ( u i - u b , i ) F i j ρ σ ( u j - u b , j ) + ( u i - u b , i ) M i j ρ σ ( u j - u b , j ) + ½ ( u i - u b , i ) B i j ρ σ ( u j - u b , j ) + ( u i - u b , i ) ( u j - u b , j ) ( u k - u b , k ) F i j k ρ σ + ½ ( u i - u b , i ) ( u j - u b , j ) ( u k - u b , k ) L i j k ρ σ + ½ ( u i - u b , i ) ( u j - u b , j ) ( u k - u b , k ) N i j k ρ σ + ( u i - u b , i ) ( u j - u b , j ) ( u k - u b , k ) B i j k ρ σ + O ( 4 ) ,
f ρ σ = { y b , ρ = 1 - p b , ρ = 0 ,             b ρ σ = { - y b σ = 1 p b , σ = 0 .
u ¯ = ( - 1 ) ρ ¯ C ρ σ ( u , u ) u ,
u ¯ = ( - 1 ) σ C ρ σ ( u , u ) u ,
y k = - [ y i M i k 01 + ( p i - p b , i ) B i k 01 + ½ y i y j L i j k 01 + y i ( p j - p b , j ) N i j k 01 + ½ ( p i - p b , i ) ( p j - p b , j ) B i j k 01 ] + O ( 3 ) .
C ρ ¯ σ ( u ¯ , u ) C ρ σ [ u ( u ¯ , u ) , u ] - ( - 1 ) ρ ¯ u ( u ¯ , u ) · u ¯ ,
C ρ ¯ σ ( u , u ¯ ) C ρ σ [ u , u ( u , u ¯ ) ] - ( - 1 ) σ u ( u , u ¯ ) · u ¯ ,
u [ C ρ σ ( u , u ) - ( - 1 ) ρ ¯ u · u ¯ ] = 0 ,
u [ C ρ σ ( u , u ) - ( - 1 ) σ u · u ¯ ] = 0 ,
F i j k ρ ¯ σ = ( - 1 ) ρ ¯ { ( F - 1 ) l i ( F - 1 ) m j ( F - 1 ) p k F l m p } ρ σ ,
L i j k ρ ¯ σ = { ( F - 1 ) l i ( F - 1 ) m i [ L l m k - ( F - 1 M ) k p F l m p ] } ρ σ ,
N i j k ρ ¯ σ = ( - 1 ) ρ ¯ { ( F - 1 ) l i [ N l j k - ( F - 1 M ) m j L l m k - ( F - 1 M ) m k L l m j + ( F - 1 M ) m j ( F - 1 M ) p k F l m p ] } ρ σ ,
B i j k ρ ¯ σ = { B i j k - ( F - 1 M ) l i N l j k - ( F - 1 M ) l j N l i k - ( F - 1 M ) l k N l i j + ( F - 1 M ) l i ( F - 1 M ) m j L l m k + ( F - 1 M ) l i ( F - 1 M ) m k L l m j + ( F - 1 M ) l j ( F - 1 M ) m k L l m i - ( F - 1 M ) l i ( F - 1 M ) m j ( F - 1 M ) p k F l m p } ρ σ ,
F i j k ρ σ ¯ = { F i j k - ( M B - 1 ) i l L j k l - ( M B - 1 ) j l L i k l - ( M B - 1 ) k l L i j l + ( M B - 1 ) i l ( M B - 1 ) j m N k l m + ( M B - 1 ) i l ( M B - 1 ) k m N j l m + ( M B - 1 ) j l ( M B - 1 ) k m N i l m - ( M B - 1 ) i l ( M B - 1 ) j m ( M B - 1 ) k p B l m p } ρ σ ,
L i j k ρ σ ¯ = ( - 1 ) σ { ( B - 1 ) k p [ L i j p - ( M B - 1 ) j m N i m p - ( M B - 1 ) i m N j m p + ( M B - 1 ) i l ( M B - 1 ) j m B l m p ] } ρ σ ,
N i j k ρ σ ¯ = { ( B - 1 ) j m ( B - 1 ) k p [ N i m p - ( M B - 1 ) i l B l m p ] } ρ σ ,
B i j k ρ σ ¯ = ( - 1 ) σ { ( B - 1 ) i l ( B - 1 ) j m ( B - 1 ) k p B l m p } ρ σ ,
C I II ρ τ ( u , u ) = C I ρ σ [ u , u ( u , u ) ] + C II ρ σ [ u ( u , u ) , u ] ,
C I ρ σ ( u , u ) u = - C II σ τ ( u , u ) u .
F i j k I II = F i j k I - M ^ i l I L j k l I - M ^ j l I L i k l I - M ^ k l I L i j l I + M ^ i l I M ^ j m I N k l m I + M ^ i l I M ^ k m I N j l m I + M ^ j l I M ^ k m I N i l m I - M ^ i l I M ^ j m I M ^ k p I ( B l m p I + F l m p II ) ,
L i j k I II = - M ^ p k II L i j p I + M ^ i l I M ^ m k II N j l m I + M ^ j l I M ^ m k II N i l m I + M ^ i l I M ^ j m I L l m k II - M ^ i l I M ^ j m I M ^ p k II ( B l m p I + F l m p II ) ,
N i j k I II = - M ^ i l II N l j k II + M ^ i l I M ^ m j II L l m k II + M ^ i l I M ^ m k II L l m j II + M ^ m j II M ^ p k II N i m p I - M ^ i l I M ^ j m II M ^ p k II ( B l m p I + F l m p II ) ,
B i j k I II = B i j k II - M ^ l i II N l j k II - M ^ l j II N l i k II - M ^ l k II N l i j II + M ^ l i II M ^ m j II L l m k II + M ^ l i II M ^ m k II L l m j II + M ^ l j II M ^ m k II L l m i II - M ^ l i II M ^ m j II M ^ p k II ( B l m p I + F l m p II ) ,
M ^ i j I : = [ M I ( B I + F II ) - 1 ] i j ,
M ^ i j II : = [ ( B I + F II ) - 1 M II ] i j ,
y = R ( φ ) A ( θ ) y ˜ - y ˜ · a ( θ ) n × [ F R ( φ ) A ( θ ) y ˜ + M R ( φ ) A - 1 ( θ ) ( p ˜ - p ˜ b ) ] + O ( 3 ) ,
p = R ( φ ) A - 1 ( θ ) ( p ˜ - p ˜ b ) - tan ( θ ) 2 n ( p ˜ - p ˜ b ) T A - 2 ( θ ) ( p ˜ - p ˜ b ) [ cos ( φ ) sin ( φ ) ] + O ( 3 ) ,
p ˜ b = n a ( θ ) ,
a ( θ ) : = [ sin ( θ ) 0 ] ,
A ( θ ) : = [ cos ( θ ) 0 0 1 ] ,
R ( φ ) : = [ cos ( φ ) - sin ( φ ) sin ( φ ) cos ( φ ) ] .
n y ˜ · a ( θ ) - y ˜ · a ( θ ) 2 n × [ F R ( φ ) A ( θ ) y ˜ + M R ( φ ) A - 1 ( θ ) ( p ˜ - p ˜ b ) ] 2 + O ( 4 ) .
F ˜ = A ( θ ) R T ( φ ) F R ( φ ) A ( θ ) ,
M ˜ = A ( θ ) R T ( φ ) M R ( φ ) A - 1 ( θ ) ,
B ˜ = A - 1 ( θ ) R T ( φ ) B R ( φ ) A - 1 ( θ ) ,
F ˜ i j k = [ R ( θ ) A ( θ ) ] l i [ R ( φ ) A ( θ ) ] m j [ R ( φ ) A ( θ ) ] p k × ( F , l m p - 1 n { ( F 2 ) l m [ R ( φ ) A - 1 ( θ ) a ( θ ) ] p + ( F 2 ) l p [ R ( φ ) A - 1 ( θ ) a ( θ ) ] m + ( F 2 ) m p [ R ( φ ) A - 1 ( θ ) a ( θ ) ] l } ) ,
L ˜ i j k = [ R ( φ ) A ( θ ) ] l i [ R ( φ ) A ( θ ) ] m j × ( L , l m p - 1 n { ( F M ) m p [ R ( φ ) A - 1 ( θ ) a ( θ ) ] l + ( F M ) l p [ R ( φ ) A - 1 ( θ ) a ( θ ) ] m } ) [ R ( φ ) A - 1 ( θ ) ] p k ,
N ˜ i j k = [ R ( φ ) A ( θ ) ] l i × { N , l m p - 1 n ( M T M ) m p [ R ( φ ) A - 1 ( θ ) a ( θ ) ] l - 1 n δ m p [ M R ( φ ) A - 1 ( θ ) a ( θ ) ] l } × [ R ( φ ) A - 1 ( θ ) ] m j [ R ( φ ) A - 1 ( θ ) ] p k ,
B ˜ i j k = ( B , l m p - 1 n { δ l m [ B R ( φ ) A - 1 ( θ ) a ( θ ) ] p + δ l p [ B R ( φ ) A - 1 ( θ ) a ( θ ) ] m + δ m p [ B R ( φ ) A - 1 ( θ ) a ( θ ) ] l } ) [ R ( φ ) A - 1 ( θ ) ] l i × [ R ( φ ) A - 1 ( θ ) ] m j [ R ( φ ) A - 1 ( θ ) ] p k ,
δ i j : = { 1 i = j 0 i j .
( N , 111 - N , 122 ) = 1 n tan ( θ ) cos ( φ ) ( M , 11 2 + M , 21 2 - M , 12 2 - M , 22 2 ) ,
( N , 111 + N , 122 ) = 1 n tan ( θ ) cos ( φ ) M 2 + 2 n tan ( θ ) × [ cos ( φ ) M , 11 + sin ( φ ) M , 12 ] ,
( N , 211 - N , 222 ) = 1 n tan ( θ ) sin ( φ ) ( M , 11 2 + M , 21 2 - M , 12 2 - M , 22 2 ) ,
( N , 211 + N , 222 ) = 1 n tan ( θ ) sin ( φ ) M 2 + 2 n tan ( θ ) × [ cos ( φ ) M , 21 + sin ( φ ) M , 22 ] ,
N , 112 = 1 n tan ( θ ) cos ( φ ) ( M , 11 M , 12 + M , 21 M , 22 ) ,
N , 212 = 1 n tan ( θ ) sin ( φ ) ( M , 11 M , 12 + M , 21 M , 22 ) ,
U : = ( U 11 2 + U 12 2 + U 21 2 + U 22 2 ) 1 / 2 .
( N , 111 - N , 122 ) ( M , 11 M , 12 + M , 21 M , 22 ) = N , 112 ( M , 11 2 + M , 21 2 - M , 12 2 - M , 22 2 ) ,
( N , 211 - N , 222 ) ( M , 11 M , 12 + M , 21 M , 22 ) = N , 212 ( M , 11 2 + M , 21 2 - M , 12 2 - M , 22 2 ) .
tan ( φ ) = N , 211 - N , 222 N , 111 - N , 122 ,
tan ( θ ) = { n ( N , 111 - N , 122 ) cos ( φ ) ( M , 11 2 + M , 21 2 - M , 12 2 - M , 22 2 ) cos ( φ ) 0 n ( N , 211 - N , 222 ) sin ( φ ) ( M , 11 2 + M , 21 2 - M , 12 2 - M , 22 2 ) sin ( φ ) 0 .
tan ( φ ) = - v 1 · m 1 v 1 · m 2 ,
tan ( θ ) = { - n v 1 · m 2 cos ( φ ) det ( M ) ( m 2 2 - m 1 2 ) cos ( φ ) 0 n v 1 · m 1 sin ( φ ) det ( M ) ( m 2 2 - m 1 2 ) sin ( φ ) 0 ,
m i : = ( M , 1 i M , 2 i ) ,
v 1 : = m 1 2 ( - N , 222 N , 122 ) - m 2 2 ( - N , 211 N , 111 ) .
sin ( φ ) = sin ( φ ) = 0 ,
tan ( θ ) = n cos ( φ ) ( N , 111 - N , 122 ) ( M , 11 2 - M , 22 2 ) ,
tan ( θ ) = n cos ( φ ) ( M , 11 2 N , 122 - M , 22 2 N , 111 ) M , 11 ( M , 11 2 - M , 22 2 ) .
tan ( φ ) = N , 212 N , 112 ,
tan ( θ ) = { n N , 112 cos ( φ ) ( M , 11 M , 12 + M , 21 M , 22 ) cos ( φ ) 0 n N , 212 sin ( φ ) ( M , 11 M , 12 + M , 21 M , 22 ) sin ( φ ) 0 ,
tan ( φ ) = - v 2 · m 1 v 2 · m 2 ,
tan ( θ ) = { - n v 2 · m 2 cos ( φ ) det ( M ) m 1 · m 2 cos ( φ ) 0 n v 2 · m 1 sin ( φ ) det ( M ) m 1 · m 2 sin ( φ ) 0 ,
v 2 : = m 1 2 ( - N , 212 N , 112 ) - ( m 1 · m 2 ) ( - N , 212 N , 111 ) .
tan ( φ ) = ( - 1 ) r tan ( φ ) ,
tan ( θ ) = - μ [ cos ( φ ) cos ( φ ) ] M , 11 tan ( θ ) ,
x s = s ( y s ) ,
x = s 0 ( y ) ,
x = s 1 ( y ) .
C 00 ( y , y ) = n 0 { [ d + cos ( θ ) s 1 ( y ) + sin ( θ ) y 1 - cos ( θ ) s 0 ( y ) - sin ( θ ) y 1 ] 2 + [ cos ( φ ) cos ( θ ) y 1 + cos ( φ ) sin ( θ ) s 0 ( y ) + sin ( φ ) y 2 - sin ( θ ) s 1 ( y ) - cos ( θ ) y 1 ] 2 + [ y 2 + sin ( φ ) sin ( θ ) s 0 ( y ) + sin ( φ ) cos ( θ ) y 1 - cos ( θ ) y 2 ] 2 } 1 / 2 .
F i j 00 = n 0 [ 1 d A 2 ( θ ) - cos ( θ ) S 0 ] i j ,
M i j 00 = - n 0 d [ A ( θ ) R ( φ ) A ( θ ) ] i j ,
B i j 00 = n 0 [ 1 d A 2 ( θ ) - cos ( θ ) S 1 ] i j ,
F i j k 00 = - 1 d { F i j 00 [ a ( θ ) ] k + F i k 00 [ a ( θ ) ] j + F j k 00 [ a ( θ ) ] i } - n 0 cos ( θ ) S 0 , i j k ,
L i j k 00 = n 0 d 2 [ A 2 ( θ ) ] i j [ a ( θ ) ] k - n 0 d S 0 , i j [ R ( φ ) A ( θ ) ] l k [ a ( θ ) ] l - 1 d M i k 00 [ a ( θ ) ] j - 1 d M j k 00 [ a ( θ ) ] i ,
N i j k 00 = - n 0 d 2 [ A 2 ( θ ) ] j k [ a ( θ ) ] i - n 0 d S 1 , j k [ A ( θ ) R ( φ ) ] i l [ a ( θ ) ] l + 1 d M i j 00 [ a ( θ ) ] k + 1 d M j k 00 [ a ( θ ) ] j ,
B i j k 00 = 1 d { B i j 00 [ a ( θ ) ] k + B i k 00 [ a ( θ ) ] j + B j k 00 [ a ( θ ) ] i } + n 0 cos ( θ ) S 1 , i j k ,
S 0 = [ S 0 , 11 S 0 , 12 S 0 , 12 S 0 , 22 ] : = [ 2 s 0 / y 1 2 2 s 0 / y 1 y 2 2 s 0 / y 1 y 2 2 s 0 / y 2 2 ] | y = 0 ,
S 0 , i j k = 3 s 0 y i y j y k | y = 0 ,
F i j 00 = n t δ i j ,             M i j 00 = - n t [ A ( θ s ) ] i j , B i j 00 = n [ 1 t A 2 ( θ s ) + cos ( θ s ) S ] i j ,
F i j k 00 = 0 ,
L i j k 00 = n t 2 δ i j [ a ( θ s ) ] k ,
N i j k 00 = - n t S j k [ a ( θ s ) ] i - n t 2 { [ A ( θ s ) ] i j [ a ( θ s ) ] k + [ A ( θ s ) ] i k [ a ( θ s ) ] j } ,
B i j k 00 = n t { [ 1 t A 2 ( θ s ) + cos ( θ s ) S ] i j [ a ( θ s ) ] k + [ 1 t A 2 ( θ s ) + cos ( θ s ) S ] i k [ a ( θ s ) ] j + [ 1 t A 2 ( θ s ) + cos ( θ s ) S ] j k [ a ( θ s ) i ] } + n cos ( θ s ) S i j k ,
F i j 01 = - n cos ( θ s ) S i j ,             M i j 01 = - [ A ( θ s ) ] i j , B i j 01 = - 1 n t δ i j ,
F i j k 01 = - n cos ( θ s ) S i j k ,
L i j k 01 = - S i j [ a ( θ s ) ] k ,
N i j k 01 = - 1 n δ j k [ a ( θ s ) ] i ,
B i j k 01 = 0.
B 01 = - t n I - t n A ( θ s ) × { A 2 ( θ s ) + t [ cos ( θ s ) - μ cos ( θ s ) ] S } - 1 A ( θ s ) ,
μ : = n / n .
S 12 = 0 ,
S 11 = cos 2 ( θ s ) S 22 + [ cos 2 ( θ s ) - cos 2 ( θ s ) ] t [ cos ( θ s ) - μ cos ( θ s ) ] ,
t = - μ t 1 + t [ cos ( θ s ) - μ cos ( θ s ) ] S 22 .
B i j k 01 = t { μ 2 - 1 - t [ cos ( θ s ) - μ cos ( θ s ) ] S 22 } n 2 cos ( θ s ) { 1 + t [ cos ( θ s ) - μ cos ( θ s ) ] S 22 } 3 × ( D i j a k + D i k a j + D j k a i ) + t 3 [ cos ( θ s ) - μ cos ( θ s ) ] n 2 { 1 + t [ cos ( θ s ) - μ cos ( θ s ) ] S 22 } 3 × ( A - 1 ) i l ( A - 1 ) j m ( A - 1 ) k p S l m p ,
D : = t cos ( θ s ) S 22 I + { cos ( θ s ) [ cos ( θ s ) - μ cos ( θ s ) ] cos ( θ s ) [ cos ( θ s ) - μ cos ( θ s ) ] 0 0 1 } .
S 111 μ 2 t 2 [ cos ( θ s ) - μ cos ( θ s ) ] 2 = 3 sin ( θ s ) cos ( θ s ) cos ( θ s ) × { 1 - μ + t [ cos ( θ s ) - μ cos ( θ s ) ] S 22 } × { [ cos ( θ s ) + μ cos ( θ s ) ] + t cos ( θ ) × [ cos ( θ s ) - μ cos ( θ s ) ] S 22 } ,
S 112 = S 222 = 0 ,
S 122 μ 2 t 2 [ cos ( θ s ) - μ cos ( θ s ) ] = sin ( θ s ) [ 1 + t cos ( θ s ) S 22 ] × { 1 - μ 2 + t [ cos ( θ s ) - μ cos ( θ s ) ] S 22 } .
S 122 t [ cos ( θ s ) - μ cos ( θ s ) ] = S 22 cos ( θ s ) sin ( θ s ) { 1 - μ 2 + t [ cos ( θ s ) - μ cos ( θ s ) ] S 22 } .
S 22 = - 1 t [ cos ( θ s ) + μ cos ( θ s ) ] .
S 11 = - 1 t { cos ( θ s ) + μ cos ( θ s ) ] ,
t = - t μ .
M 01 = - 1 μ 2 [ cos ( θ s ) / cos ( θ s ) 0 0 1 ] ,
N 111 01 = ( μ 2 - 1 ) sin ( θ s ) n μ 4 cos 3 ( θ s ) ,
N 122 01 = N 111 01 cos 2 ( θ s ) .
φ = φ = π ,
θ = θ s ,
θ = θ s .
- M ˜ 11 = - M ˜ 22 = - 1 / μ 2 .
L ˜ 111 = L ˜ 221 = ( 1 - μ 2 ) sin ( θ s ) t μ 4 cos ( θ s ) cos ( θ s ) ,
F 01 = n [ t I + n n cos ( θ s ) - n cos ( θ s ) A S - 1 A ] - 1 ,
M 01 = - A { A 2 + t n [ n cos ( θ s ) - n cos ( θ s ) ] S } - 1 A ,
B 01 = - t n I - t n A { A 2 + t n [ n cos ( θ s ) - n cos ( θ s ) ] S } - 1 A ,
F i j k 01 = n 3 [ n cos ( θ s ) - n cos ( θ s ) ] ( A G ) i l ( A G ) j m ( A G ) k p S l m p + { n 3 t [ ( A G S G A ) i j U k l + ( A G S G A ) i k U j l + ( A G S G A ) j k U i l ] + n 2 t 2 [ ( A G ) i l T j k + ( A G ) j l T i k + ( A G ) k l T i j ] } a l ,
L i j k 01 = t n 2 [ n cos ( θ s ) - n cos ( θ s ) ] ( A G ) i l ( A G ) j m ( A G ) k p S l m p + n t T i j ( A G ) k l a l + [ n 2 ( A G S G A ) i k U j l + n 2 ( A G S G A ) j k U i l + n 2 n ( A G S G A ) i j V k l ] a l ,
N i j k 01 = t 2 n [ n cos ( θ s ) - n cos ( θ s ) ] ( A G ) i l ( A G ) j m ( A G ) k p S l m p - n n 2 W j k ( A G ) i l a l + [ t n n ( A G S G A ) i j V k l + t n n ( A G S G A ) i k V j l + t n ( A G S G A ) j k U i l ] a l ,
B i j k 01 = t 3 [ n cos ( θ s ) - n cos ( θ s ) ] ( A G ) i l ( A G ) j m ( A G ) k p S l m p + { t 2 n [ ( A G S G A ) i j V k l + ( A G S G A ) i k V j l + ( A G S G A ) j k V i l ] - t n 2 [ ( A G ) i l W j k + ( A G ) j l W i k + ( A G ) k l W i j ] } a l ,
G : = { n A 2 + t [ n cos ( θ s ) - n cos ( θ s ) ] S } - 1 .
T : = I - n ( A G A ) ,
U : = n cos ( θ s ) ( A G ) - I ,
V : = n 2 cos ( θ s ) ( A G ) - n I ,
W : = n I - n 2 ( A G A ) .
F i j 10 = - t n δ i j ,             M i j 01 = ( A ) i j ,             B i j 10 = n cos ( θ s ) S i j ,
F i j k 10 = 0 ,
L i j k 10 = 1 n δ i j a k ,
N i j k 10 = a i S j k ,
B i j k 10 = n cos ( θ s ) S i j k .
F 11 = - t n I - 1 n cos ( θ s ) - n cos ( θ s ) ( A S - 1 A ) ,
M 11 = 1 n cos ( θ s ) - n cos ( θ s ) ( A S - 1 A ) ,
B 11 = - t n I - 1 n cos ( θ s ) - n cos ( θ s ) ( A S - 1 A ) ,
F i j k 11 = 1 [ n cos ( θ s ) - n cos ( θ s ) ] 2 [ ( A S - 1 A ) i j a k + ( A S - 1 A ) i k a j + ( A S - 1 A ) j k a i - ( A S - 1 ) i l ( A S - 1 ) j m ( A S - 1 ) k p S l m p ] - 1 n [ n cos ( θ s ) - n cos ( θ s ) ] [ ( A S - 1 ) i l δ j k + ( A S - 1 ) j l δ i k + ( A S - 1 ) k l δ i j ] a l ,
L i j k 11 = 1 [ n cos ( θ s ) - n cos ( θ s ) ] 2 [ - n n ( A S - 1 A ) i j a k - ( A S - 1 A ) i k a j - ( A S - 1 A ) j k a i + ( A S - 1 ) i l ( A S - 1 ) j m ( A S - 1 ) k p S l m p ] + 1 n [ n cos ( θ s ) - n cos ( θ s ) ] [ ( A S - 1 ) k l a l δ i j ] ,
N i j k 11 = 1 [ n cos ( θ s ) - n cos ( θ s ) ] 2 [ n n ( A S - 1 A ) i j a k + n n ( A S - 1 A ) i k a j + ( A S - 1 A ) j k a i - ( A S - 1 ) i l ( A S - 1 ) j m ( A S - 1 ) k p S l m p ] + 1 n 2 [ n cos ( θ s ) - n cos ( θ s ) ] [ ( A S - 1 ) k l a l δ j k ] ,
B i j k 11 = - n n [ n cos ( θ s ) - n cos ( θ s ) ] 2 [ ( A S - 1 A ) i j a k + ( A S - 1 A ) i k a j + ( A S - 1 A ) j k a i - n n ( A S - 1 ) i l ( A S - 1 ) j m ( A S - 1 ) k p S l m p ] + n n 2 [ n cos ( θ s ) - n cos ( θ s ) ] [ ( A S - 1 ) i l δ j k + ( A S - 1 ) j l δ i k + ( A S - 1 ) k l δ i j ] a l .
x = s ( y ) ,
x = s ( y ) .
F = F - n cos ( θ ) S + n cos ( θ ) M S M ,
M = M ,
B = [ 0 0 0 0 ] ,
F i j k = F i j k - n cos ( θ ) S i j k - n cos ( θ ) M i l M j m M k p S l m p + i j k + i k j + j k i + n 3 cos 3 ( θ ) ( M S ) i l ( M S ) j m ( M S ) k p B l m p ,
L i j k = L i j k + 1 cos ( θ ) [ M T a ( θ ) k S i j + 1 cos ( θ ) [ M S M T ] i j [ a ( θ ) ] k + n cos ( θ ) [ ( M S ) i l N j k l + ( M S ) j l N i k l ] + n 2 cos 2 ( θ ) ( M S ) i l ( M S ) j m B l m k ,
N i j k = N i j k + n cos ( θ ) ( M S ) i l B l j k ,
B i j k = B i j k ,
i j k : = n cos ( θ ) ( M S ) k l L i j l + n 2 cos 2 ( θ ) ( M S ) i l ( M S ) j m N k l m + n ( M S M T ) i j [ M S a ( θ ) ] k .
S22=-1t[1cos(θs)-μcos(θs)].

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