Abstract

A formula is derived for the illuminance at any surface in an optical system. By tracing a single ray one can compute the flux density at the image plane or any other position along the ray. The formula involves the ratio of the products of the principal curvatures of the wave front as it approaches each surface to products of the same quantities after the wave front is refracted at each surface. A procedure is presented for determining the required principal curvatures by generalizing the Coddington equations to multiple surfaces for both meridional and skew rays. Results are applicable to both spherical and aspherical surfaces. Since principal radii of curvature specify points on the caustic surfaces, the formula and computation procedure automatically yields the equations for caustic surfaces as a by-product. To illustrate the computation procedure the illuminance and caustic surfaces are derived for an aspherical singlet.

© 1981 Optical Society of America

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References

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  1. D. L. Shealy, D. G. Burkhard, Opt. Acta. 22, 485 (1975).
    [CrossRef]
  2. D. L. Shealy, D. G. Burkhard, Opt. Acta. 20, 287 (1973).
    [CrossRef]
  3. D. G. Burkhard, D. L. Shealy, J. Opt. Soc. Am. 63, 299 (1973).
    [CrossRef]
  4. D. L. Shealy, D. G. Burkhard, Appl. Opt. 12, 2955 (1973).
    [CrossRef] [PubMed]
  5. D. G. Burkhard, D. L. Shealy, Int. J. Heat Mass Transfer 16, 1492 (1973).
    [CrossRef]
  6. E. Bochove, J. Opt. Soc. Am. 69, 891 (1979).
    [CrossRef]
  7. M. Herzberger, Modern Geometrical Optics, Pure and Applied Mathematical Series (Wiley-Interscience, New York, 1958).
  8. O. N. Stavroudis, D. P. Feder, J. Opt. Soc. Am. 44, 163 (1954).
    [CrossRef]
  9. J. B. Keller, H. B. Keller, J. Opt. Soc. Am. 40, 48 (1950).
    [CrossRef]
  10. V. A. Fock, Electromagnetic Diffraction and Propagation Problems (Pergamon, New York, 1965).
  11. G. A. Deschamps, Proc. IEEE 60.9, 1022 (1972).
    [CrossRef]
  12. S. W. Lee, IEEE Trans. Antennas Propag. AP-23, 184 (1975).
  13. M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1975), p. 131.
  14. D. G. Burkhard, Appl. Opt. 19, 3676 (1980).
    [CrossRef]
  15. C. E. Weatherburn, Differential Geometry of Three Dimensions (Cambridge, V.P., New York, 1931), Vol. 1, p. 66.
  16. O. N. Stavroudis, R. C. Fronczek, J. Opt. Soc. Am. 66, 795 (1976).
    [CrossRef]
  17. S. C. Parker, “Properties and Applications of Generalized Ray Tracing,” Technical Report 71, Optical Sciences Center (U. Arizona, Tucson, 1971).
  18. J. N. Kneisly, J. Opt. Soc. Am. 54, 229 (1964).
    [CrossRef]
  19. O. N. Stavroudis, The Optics of Rays, Wave fronts, and Caustics, (Academic, New York, 1972), Chap. 10.
  20. O. N. Stavroudis, J. Opt. Soc. Am. 66, 1330 (1976).
    [CrossRef]
  21. A. E. Murray, J. Opt. Soc. Am. 47, 599 (1957).
    [CrossRef]
  22. J. C. Sturm, J. Math, pur et Appl. 3, 357 (1838).
  23. E. Kreyszig, Introduction to Differential Geometry, Math. Exp. 16 (U. Toronto Press, Ontario, 1968), pp. 38, 82.
  24. F. A. Jenkins, H. E. White, Fundamentals of Optics (McGraw-Hill, New York, 1950), pp. 95, 149.
  25. Military Standardization Handbook—Optical Design, MIL-HDBK-141 (Defense Supply Agency, Washington, D.C., 1962), Chap. 5, pp. 14–20.
  26. Ref. 23, p. 95.
  27. Ref. 23, p. 93.
  28. Ref. 23, pp. 91–92.
  29. M. M. Lipschutz, Differential Geometry (McGraw-Hill, New York, 1969), p. 196.

1980 (1)

1979 (1)

1976 (2)

1975 (2)

S. W. Lee, IEEE Trans. Antennas Propag. AP-23, 184 (1975).

D. L. Shealy, D. G. Burkhard, Opt. Acta. 22, 485 (1975).
[CrossRef]

1973 (4)

D. L. Shealy, D. G. Burkhard, Opt. Acta. 20, 287 (1973).
[CrossRef]

D. G. Burkhard, D. L. Shealy, J. Opt. Soc. Am. 63, 299 (1973).
[CrossRef]

D. L. Shealy, D. G. Burkhard, Appl. Opt. 12, 2955 (1973).
[CrossRef] [PubMed]

D. G. Burkhard, D. L. Shealy, Int. J. Heat Mass Transfer 16, 1492 (1973).
[CrossRef]

1972 (1)

G. A. Deschamps, Proc. IEEE 60.9, 1022 (1972).
[CrossRef]

1964 (1)

1957 (1)

1954 (1)

1950 (1)

1838 (1)

J. C. Sturm, J. Math, pur et Appl. 3, 357 (1838).

Bochove, E.

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1975), p. 131.

Burkhard, D. G.

D. G. Burkhard, Appl. Opt. 19, 3676 (1980).
[CrossRef]

D. L. Shealy, D. G. Burkhard, Opt. Acta. 22, 485 (1975).
[CrossRef]

D. L. Shealy, D. G. Burkhard, Appl. Opt. 12, 2955 (1973).
[CrossRef] [PubMed]

D. G. Burkhard, D. L. Shealy, Int. J. Heat Mass Transfer 16, 1492 (1973).
[CrossRef]

D. L. Shealy, D. G. Burkhard, Opt. Acta. 20, 287 (1973).
[CrossRef]

D. G. Burkhard, D. L. Shealy, J. Opt. Soc. Am. 63, 299 (1973).
[CrossRef]

Deschamps, G. A.

G. A. Deschamps, Proc. IEEE 60.9, 1022 (1972).
[CrossRef]

Feder, D. P.

Fock, V. A.

V. A. Fock, Electromagnetic Diffraction and Propagation Problems (Pergamon, New York, 1965).

Fronczek, R. C.

Herzberger, M.

M. Herzberger, Modern Geometrical Optics, Pure and Applied Mathematical Series (Wiley-Interscience, New York, 1958).

Jenkins, F. A.

F. A. Jenkins, H. E. White, Fundamentals of Optics (McGraw-Hill, New York, 1950), pp. 95, 149.

Keller, H. B.

Keller, J. B.

Kneisly, J. N.

Kreyszig, E.

E. Kreyszig, Introduction to Differential Geometry, Math. Exp. 16 (U. Toronto Press, Ontario, 1968), pp. 38, 82.

Lee, S. W.

S. W. Lee, IEEE Trans. Antennas Propag. AP-23, 184 (1975).

Lipschutz, M. M.

M. M. Lipschutz, Differential Geometry (McGraw-Hill, New York, 1969), p. 196.

Math, J.

J. C. Sturm, J. Math, pur et Appl. 3, 357 (1838).

Murray, A. E.

Parker, S. C.

S. C. Parker, “Properties and Applications of Generalized Ray Tracing,” Technical Report 71, Optical Sciences Center (U. Arizona, Tucson, 1971).

Shealy, D. L.

D. L. Shealy, D. G. Burkhard, Opt. Acta. 22, 485 (1975).
[CrossRef]

D. G. Burkhard, D. L. Shealy, J. Opt. Soc. Am. 63, 299 (1973).
[CrossRef]

D. L. Shealy, D. G. Burkhard, Appl. Opt. 12, 2955 (1973).
[CrossRef] [PubMed]

D. G. Burkhard, D. L. Shealy, Int. J. Heat Mass Transfer 16, 1492 (1973).
[CrossRef]

D. L. Shealy, D. G. Burkhard, Opt. Acta. 20, 287 (1973).
[CrossRef]

Stavroudis, O. N.

Sturm, J. C.

J. C. Sturm, J. Math, pur et Appl. 3, 357 (1838).

Weatherburn, C. E.

C. E. Weatherburn, Differential Geometry of Three Dimensions (Cambridge, V.P., New York, 1931), Vol. 1, p. 66.

White, H. E.

F. A. Jenkins, H. E. White, Fundamentals of Optics (McGraw-Hill, New York, 1950), pp. 95, 149.

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1975), p. 131.

Appl. (1)

J. C. Sturm, J. Math, pur et Appl. 3, 357 (1838).

Appl. Opt. (2)

IEEE Trans. Antennas Propag. (1)

S. W. Lee, IEEE Trans. Antennas Propag. AP-23, 184 (1975).

Int. J. Heat Mass Transfer (1)

D. G. Burkhard, D. L. Shealy, Int. J. Heat Mass Transfer 16, 1492 (1973).
[CrossRef]

J. Opt. Soc. Am. (8)

Opt. Acta. (2)

D. L. Shealy, D. G. Burkhard, Opt. Acta. 22, 485 (1975).
[CrossRef]

D. L. Shealy, D. G. Burkhard, Opt. Acta. 20, 287 (1973).
[CrossRef]

Proc. IEEE (1)

G. A. Deschamps, Proc. IEEE 60.9, 1022 (1972).
[CrossRef]

Other (13)

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1975), p. 131.

S. C. Parker, “Properties and Applications of Generalized Ray Tracing,” Technical Report 71, Optical Sciences Center (U. Arizona, Tucson, 1971).

C. E. Weatherburn, Differential Geometry of Three Dimensions (Cambridge, V.P., New York, 1931), Vol. 1, p. 66.

V. A. Fock, Electromagnetic Diffraction and Propagation Problems (Pergamon, New York, 1965).

M. Herzberger, Modern Geometrical Optics, Pure and Applied Mathematical Series (Wiley-Interscience, New York, 1958).

O. N. Stavroudis, The Optics of Rays, Wave fronts, and Caustics, (Academic, New York, 1972), Chap. 10.

E. Kreyszig, Introduction to Differential Geometry, Math. Exp. 16 (U. Toronto Press, Ontario, 1968), pp. 38, 82.

F. A. Jenkins, H. E. White, Fundamentals of Optics (McGraw-Hill, New York, 1950), pp. 95, 149.

Military Standardization Handbook—Optical Design, MIL-HDBK-141 (Defense Supply Agency, Washington, D.C., 1962), Chap. 5, pp. 14–20.

Ref. 23, p. 95.

Ref. 23, p. 93.

Ref. 23, pp. 91–92.

M. M. Lipschutz, Differential Geometry (McGraw-Hill, New York, 1969), p. 196.

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Figures (3)

Fig. 1
Fig. 1

Origin O; source and first refracting element dS(1). Angle of incidence ϕ(i) and angle of refraction ϕ(s) are labeled.

Fig. 2
Fig. 2

Element of wave front area after emerging from S2·r2c and r 2 c is the principal radii of curvature. After focusing at a point on each of the two caustic surfaces, the wave front diverges, and an element of area on the wave when it reaches S3 is represented by dS3. Principal radii of curvature at that stage are r2p and r 2 p.

Fig. 3
Fig. 3

Arbitrary refracting surface S, unit vector Â(i) along incident ray, Â(s) along refracted ray. Â(i), x1(i), and x2(i) are orthogonal axes associated with incident wave front. Â(i) lies along the normal to the incident wave front. Â(s), x1(s), and x2(s) are corresponding quantities on refracted wave front. N ˆ is the unit normal to the surface, and x1 and x2 are orthogonal vectors in the plane tangent to the surface. x2(i), x2, and x2(s) all lie in the plane of incidence. The angle between x2(i) and x2 is ϕ(i). The angle between x2 and x2(s) is ϕ(s). x1(i), x1, and x2(s) are parallel and lie in the perpendicular plane, that is, the plane orthogonal to the plane of incidence. The three sets of axes have a common origin at the point of incidence of the incoming ray Â(i) but are shown separated in the figure for clarity. Â(i) and N ˆ form the plane of incidence. The intersection of the plane of incidence with the surface forms a curve whose vector curvature lies in the plane of incidence. The scalar value of this curvature vector is called the normal curvature. The plane perpendicular to Â(i) at the point of incidence intersects the surface S in a curve whose curvature we call the perpendicular curvature. Principal curvatures are intrinsic to the surface and are the extremum values of the surface curvature, one a maximum and the other a minimum. They lie, respectively, in two orthogonal planes containing N ˆ. Orientation of these planes will make some angle α with respect to the normal plane s.

Equations (168)

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X ( O ) = x p + r A ˆ ( O ) .
d S ( 1 ) cos ϕ ( 1 ) = d W ( O ) ,
X ( 1 ) = x ( u 1 , υ 1 ) + r ( 1 ) A ˆ ( u 1 , υ 1 ) ,
x ( u 1 , υ 1 ) = I x ( u 1 , υ 1 ) + J y ( u 1 , υ 1 ) + K z ( u 1 , υ 1 )
n 0 r ( 0 ) + n 1 r ( 1 ) = c 1
X ( 1 ) = x ( u 1 , υ 1 ) + [ c 1 n 0 r ( 0 ) ] n 0 A ˆ ( u 1 , υ 1 ) .
n 0 r ( 0 ) + n 1 r ( 1 ) + n 2 r ( 2 ) = c 2 ,
X ( 2 ) = x ( u 2 , υ 2 ) + [ c 2 n 0 r ( 0 ) n 0 r ( 1 ) ] n 2 A ˆ ( u 2 , υ 2 ) .
d S ( 2 ) cos ϕ ( 2 ) = d W ( 1 ) ,
X ( 2 ) = x ( u 2 , υ 2 ) + r ( 2 ) A ˆ ( u 2 , υ 2 ) .
d F = I 0 d Ω = I 0 d W ( 0 ) / r 2 ( 0 ) .
σ ( 0 ) = B cos ϕ ( 0 ) d S ( 0 ) / r 2 ( 0 ) ,
d F d S ( 0 ) d S ( 1 ) = σ ( 0 ) cos ϕ ( 1 ) d S ( 1 ) .
E d S ( 0 ) d S ( 2 ) = σ ( 0 ) ρ 1 cos ϕ ( 1 ) d S ( 1 ) d S ( 2 ) .
E d S ( 0 ) d S ( 3 ) = σ ( 0 ) ρ 1 ρ 2 cos ϕ ( 1 ) d S ( 1 ) d S ( 2 ) d S ( 2 ) d S ( 3 )
d S ( 1 ) = | x ( 1 ) u 1 × x ( 1 ) υ 1 | d u 1 d υ 1 = g 1 / 2 ( 1 ) d u 1 d υ 1 ,
g ( 1 ) = g 11 ( 1 ) g 22 ( 1 ) g 12 2 ( 1 ) , g 11 = x 1 ( 1 ) · x 1 ( 1 ) , g 22 = x 2 ( 1 ) · x 2 ( 1 ) , g 12 ( 1 ) = x 1 ( 1 ) · x 2 ( 1 ) ,
x 1 x ( 1 ) / u 1 and x 2 x ( 1 ) / υ 1 .
d S ( 1 ) d S ( 2 ) = cos ϕ ( 2 ) d S ( 1 ) d W ( 1 ) .
d W ( 1 ) = X ( 1 ) u 1 × X ( 1 ) υ 1 d u 1 d υ 1 , d W ( 1 ) = { x ( 1 ) u 1 × x ( 1 ) υ 1 + r ( 12 ) [ x ( 1 ) u 1 × A ˆ ( 1 ) υ 1 + A ˆ ( 1 ) u 1 × x ( 1 ) υ 1 ] + r 2 ( 12 ) A ˆ ( 1 ) u 1 × A ˆ ( 1 ) υ 1 } d u 1 d υ 1 ,
d W ( 1 ) = { A ˆ ( 1 ) · [ x 1 ( 1 ) × x 2 ( 1 ) ] + r ( 12 ) A ˆ ( 1 ) · [ x 1 ( 1 ) × A ˆ 2 ( 1 ) + A ˆ 1 ( 1 ) × x 2 ( 1 ) ] + r 2 ( 12 ) A ˆ ( 1 ) · [ A ˆ 1 ( 1 ) × A ˆ 2 ( 1 ) ] } d u 1 d υ 1 .
d S ( 1 ) d S ( 2 ) = cos ϕ ( 2 ) { A ˆ ( 1 ) · x 1 ( 1 ) × x 2 ( 1 ) + r ( 12 ) A ˆ ( 1 ) · [ x 1 ( 1 ) × A ˆ 2 ( 1 ) + A ˆ 1 ( 1 ) × x 2 ( 1 ) ] + r 2 ( 12 ) A ˆ ( 1 ) · A ˆ 1 ( 1 ) × A ˆ 2 ( 1 ) } / g 1 / 2 ( 1 ) .
d S ( 1 ) d S ( 2 ) = cos ϕ ( 2 ) L ( 1 ) ,
L ( 1 ) = L 0 ( 1 ) + r ( 12 ) L 1 ( 1 ) + r 2 ( 12 ) L 2 ( 1 )
L 0 ( 1 ) = A ˆ ( 1 ) · [ x 1 ( 1 ) × x 2 ( 1 ) ] / g 1 / 2 ( 1 ) = cos ϕ s ( 1 ) ,
L 1 ( 1 ) = A ˆ ( 1 ) · [ x 1 ( 1 ) × A ˆ 2 ( 1 ) + A ˆ 1 ( 1 ) × x 2 ( 1 ) ] / g 1 / 2 ( 1 ) ,
L 2 ( 1 ) = A ˆ ( 1 ) · [ A ˆ 1 ( 1 ) × A ˆ 2 ( 1 ) ] / g 1 / 2 ( 1 ) .
d S ( 2 ) d S ( 3 ) = cos ϕ ( 3 ) L ( 2 ) ,
L ( 2 ) = L 0 ( 2 ) + r ( 23 ) L 1 ( 2 ) + r 2 ( 23 ) L 2 ( 2 ) ,
L 0 ( 2 ) = A ˆ ( 2 ) · [ x 1 ( 2 ) × x 2 ( 2 ) ] / g 1 / 2 ( 2 ) = cos ϕ s ( 2 ) ,
L 1 ( 2 ) = A ˆ ( 2 ) · [ x 1 ( 2 ) × A ˆ 2 ( 2 ) + A ˆ 1 ( 2 ) × x 2 ( 2 ) ] / g 1 / 2 ( 2 ) ,
L 2 ( 2 ) = A ˆ ( 2 ) · [ A ˆ 1 ( 2 ) × A ˆ 2 ( 2 ) ] / g 1 / 2 ( 2 ) .
E d S ( 0 ) d S ( 3 ) = σ ( 0 ) ρ 1 ρ 2 cos ϕ ( 1 ) cos ϕ ( 2 ) cos ϕ ( 3 ) L ( 1 ) L ( 2 ) ,
L ( 2 ) = L 0 ( 2 ) + r ( 23 ) L 1 ( 2 ) + r 2 ( 23 ) L 2 ( 2 ) = 0
x ( 2 c ) = x ( 2 ) + r c ( 23 ) A ˆ ( 2 ) ,
L ( 2 ) = cos ϕ s ( 2 ) r 2 ( 23 ) { [ 1 r 2 ( 23 ) + L 1 ( 2 ) r ( 23 ) cos ϕ s ( 2 ) + L 2 ( 2 ) cos ϕ s ( 2 ) ] } .
L ( 2 ) = cos ϕ s ( 2 ) r 2 ( 23 ) [ 1 r ( 23 ) 1 r ( 2 c ) ] [ 1 r ( 23 ) 1 r ( 2 c ) ] = cos ϕ s ( 2 ) [ 1 r ( 23 ) r ( 2 c ) ] [ 1 r ( 23 ) r ( 2 c ) ] .
r ( 23 ) = r ( 2 c ) + r ( 2 p ) = r ( 2 c ) + r ( 2 p ) .
L ( 2 ) = cos ϕ s ( 2 ) r ( 2 p ) r ( 2 c ) r ( 2 p ) r ( 2 c ) .
L ( 1 ) = cos ϕ s ( 1 ) r ( 1 p ) r ( 1 c ) r ( 1 p ) r ( 1 c ) .
E d s ( 1 ) E d s ( 3 ) = σ ( 0 ) ρ 1 ρ 2 cos ϕ ( 1 ) cos ϕ ( 2 ) cos ϕ ( 3 ) cos ϕ s ( 1 ) cos ϕ s ( 2 ) × [ r ( 1 c ) r ( 1 c ) r ( 1 p ) r ( 1 p ) ] [ r ( 2 c ) r ( 2 c ) r ( 2 p ) r ( 2 p ) ] ,
r ( 2 p ) = r ( 23 ) r ( 2 c ) ; r ( 1 p ) = r ( 12 ) r ( 1 c ) , r ( 2 p ) = r ( 23 ) r ( 2 c ) ; r ( 1 p ) = r ( 12 ) r ( 1 c ) ,
L ( 2 ) cos ϕ s ( 2 ) + r ( 12 ) L 1 ( 2 ) + r 2 ( 12 ) L 2 ( 2 ) = 0.
L ( 1 ) cos ϕ s ( 1 ) + r ( 12 ) L 1 ( 1 ) + r 2 ( 12 ) L 2 ( 1 ) = 0.
d N ˆ = N ˆ 1 d u + N ˆ 2 d υ d x = x 1 d u + x 2 d υ
( N ˆ , N ˆ 1 , d x 1 ) d u 2 + [ ( N ˆ , N ˆ 1 , x 2 ) + ( N ˆ , N ˆ 2 , x 1 ) d u d υ ] + ( N ˆ , N ˆ 2 x 2 ) d υ 2 ,
( g 11 b 12 g 12 b 11 ) d u 2 + ( g 11 b 22 g 22 b 11 ) d u d υ + ( g 12 b 22 g 22 b 12 ) d υ 2 ,
b 11 = N ˆ · x 11 , b 12 = N ˆ · x 12 , b 22 = N ˆ · x 22 , g 11 = x 1 · x 1 , g 12 = x 1 · x 2 , g 22 = x 2 · x 2 .
d S 2 cos ϕ s ( 2 ) = r 2 c r 2 c d θ 2 d θ 2 , d S 3 cos ϕ ( 3 ) = r 2 p r 2 p d θ 2 d θ 2 .
d S 1 cos ϕ s ( 1 ) = r 1 c r 1 c d θ 1 d θ 1 , d S 2 cos ϕ ( 2 ) = r 1 p r 1 p d θ 1 d θ 1 .
d S 1 d S 2 = r 1 c r 1 c r 1 p r 1 p cos ϕ ( 2 ) cos ϕ s ( 1 ) , d S 2 d S 3 = r 2 c r 2 c r 2 p r 2 p cos ϕ ( 3 ) cos ϕ s ( 3 ) .
A ˆ ( s ) = γ A ˆ ( i ) + Ω N ˆ .
γ = n ( i ) / n ( s ) and Ω = γ cos ϕ ( i ) + cos ϕ ( s ) .
γ = 1 and Ω = 2 cos ϕ ( i ) .
A ˆ ( s ) u ( s ) d u ( s ) + A ˆ ( s ) υ ( s ) d υ ( s ) γ A ˆ ( i ) u ( i ) d u ( i ) γ A ˆ ( i ) υ ( i ) d υ ( i ) d Ω N ˆ Ω N ˆ u d u Ω N ˆ υ d υ = 0
A ˆ ( s ) u ( s ) × N ˆ d u ( s ) + A ˆ ( s ) υ ( s ) × N ˆ d υ ( s ) γ A ˆ ( i ) u ( i ) × N ˆ d u ( i ) γ A ˆ ( i ) υ ( i ) × N ˆ d υ ( i ) Ω N u × N ˆ d u Ω N ˆ υ × N ˆ d υ = 0 ,
d W ( i ) = | x 1 ( i ) | | x 2 ( i ) | d u ( i ) d υ ( i ) = g 11 1 / 2 ( i ) g 22 1 / 2 ( i ) d u ( i ) d υ ( i ) ,
d S = | x 1 | | x 2 | d u d υ = g 11 1 / 2 g 22 1 / 2 d u d υ .
d W ( s ) = | x 1 ( s ) | | x 1 ( s ) | d u ( s ) d υ ( s ) = g 11 1 / 2 ( s ) g 22 1 / 2 ( s ) d u ( s ) d υ ( s ) .
d S cos ϕ ( i ) = d W ( i ) and d S cos ϕ ( s ) = d W ( s ) .
g 11 1 / 2 ( i ) d u ( i ) = g 11 1 / 2 d u = g 11 1 / 2 ( s ) d u ( s ) .
g 22 1 / 2 d υ cos ϕ ( i ) = g 22 1 / 2 ( i ) d υ ( i ) ; g 22 1 / 2 d υ cos ϕ ( s ) = g 22 1 / 2 ( s ) d υ ( s ) .
g 11 1 / 2 ( i ) = g 11 1 / 2 = g 22 1 / 2 ( s ) ; g 22 1 / 2 ( i ) cos ϕ ( i ) = g 22 1 / 2 = g 22 1 / 2 ( s ) cos ϕ ( s ) ,
N ˆ u = c u 1 x 1 + c u 2 x 2 ,
N ˆ u · x j = N ˆ · x j u ,
b u 1 = c u 1 g 11 + c u 2 g 12 b u 2 = c u 1 g 21 + c u 2 g 22 ,
N ˆ u = b 11 g 11 x 1 b 12 g 22 x 2
N ˆ υ = b 21 g 11 x 1 b 22 g 22 x 2 .
κ = b 11 / g 11 .
τ = N ˆ · d x 2 d s = N ˆ · x ˆ 2 u u s N ˆ · x ˆ 2 υ υ s = b 12 ( g 11 g 22 ) 1 / 2 ,
N ˆ u = g 11 1 / 2 κ x ˆ 1 + g 11 1 / 2 τ x ˆ 2 , N ˆ υ = g 22 1 / 2 τ x ˆ 1 g 22 1 / 2 κ x ˆ 2 .
A ˆ ( i ) u ( i ) = g 11 1 / 2 ( i ) κ x ˆ 1 ( i ) + g 11 1 / 2 ( i ) τ ( i ) x ˆ 2 ( i ) , A ˆ ( i ) υ ( i ) = g 22 1 / 2 ( i ) τ ( i ) x ˆ 1 ( i ) g 22 1 / 2 ( i ) κ ( i ) x ˆ 2 ( i ) , A ˆ ( s ) u ( s ) = g 11 1 / 2 ( s ) κ ( s ) x ˆ 1 ( s ) + g 11 1 / 2 ( s ) τ ( s ) x ˆ 2 ( s ) , A ˆ ( s ) υ ( s ) = g 22 1 / 2 ( s ) τ ( s ) x ˆ 1 ( s ) g 22 1 / 2 ( s ) κ ( s ) x ˆ 2 ( s ) .
x ˆ 1 × N ˆ = x ˆ 1 ( i ) × N ˆ = x ˆ 1 ( s ) × N ˆ = x ˆ 2
x ˆ 2 × N ˆ = x ˆ 1 ; x ˆ 2 ( i ) × N ˆ = x ˆ 1 cos ϕ ( i ) and x ˆ 2 ( s ) × N ˆ = x ˆ 1 cos ϕ ( s ) .
A ˆ ( i ) u ( i ) × N ˆ = g 11 1 / 2 ( i ) κ ( i ) x ˆ 2 + g 11 1 / 2 ( i ) τ ( i ) x ˆ 1 cos ϕ ( i ) , A ˆ ( i ) υ ( i ) × N ˆ = g 22 1 / 2 ( i ) τ ( i ) x ˆ 2 g 22 1 / 2 ( i ) κ ( i ) x ˆ 1 cos ϕ ( i ) , N ˆ u × N ˆ = g 11 1 / 2 κ x ˆ 2 + g 11 1 / 2 τ x ˆ 1 , N ˆ υ × N ˆ = g 22 1 / 2 τ x 2 g 22 1 / 2 κ x ˆ 1 , A ˆ ( s ) u ( s ) × N ˆ = g 11 1 / 2 ( s ) κ ( s ) x ˆ 2 + g 11 1 / 2 ( s ) τ ( s ) x ˆ 1 cos ϕ ( s ) , A ( s ) υ ( s ) × N ˆ = g 22 1 / 2 ( s ) τ ( s ) x ˆ 2 g 22 1 / 2 ( s ) κ ( s ) x ˆ 1 cos ϕ ( s ) .
g 11 1 / 2 d u { x ˆ 1 [ τ ( s ) cos ϕ ( s ) γ τ ( i ) cos ϕ ( i ) Ω τ ] + x ˆ 2 [ κ ( s ) γ κ ( i ) Ω κ ] } + g 22 1 / 2 d υ { x ˆ 1 [ κ ( s ) cos 2 ϕ ( s ) + γ cos 2 ϕ ( i ) κ ( i ) + Ω κ ] x ˆ 2 [ τ ( s ) cos ϕ ( s ) + γ cos ϕ ( i ) κ ( i ) + Ω τ ] } = 0.
κ ( s ) = γ κ ( i ) + Ω κ ;
κ ( s ) cos 2 ϕ ( s ) = γ κ ( i ) cos 2 ϕ ( i ) + Ω κ ;
τ ( s ) cos ϕ ( s ) = γ τ ( i ) cos ϕ ( i ) + Ω τ .
1 r ( s ) = γ r ( i ) + Ω r ,
1 r ( s ) = γ r ( i ) cos 2 ϕ ( 1 ) cos 2 ϕ ( s ) + Ω r cos 2 ϕ ( s ) ,
1 σ ( s ) = γ cos ϕ ( i ) σ ( i ) cos ϕ ( s ) + Ω σ cos ϕ ( s ) .
1 r 1 c = γ ( 1 ) ( r 01 ) + [ γ ( 1 ) cos ϕ ( 1 ) + cos ϕ s ( 1 ) ] R 1 , 1 r 1 c = γ ( 1 ) cos 2 ϕ ( 1 ) cos 2 ϕ s ( 1 ) ( r 01 ) + [ γ ( 1 ) cos ϕ ( 1 ) + cos ϕ s ( 1 ) ] R 1 cos 2 ϕ s ( 1 ) .
n 1 r 1 c + n 0 r 01 = n 1 n 0 R 1 ,
r 1 p = r 12 r 1 c , r 1 p = r 12 r 1 c .
1 r 2 c = γ ( 2 ) ( r 1 p ) + [ γ ( 2 ) cos ϕ ( 2 ) + cos ϕ s ( 2 ) ] R 2 , 1 r 2 c = γ ( 2 ) cos 2 ϕ ( 2 ) ( r 1 p ) cos 2 ϕ s ( 2 ) + [ γ ( 2 ) cos ϕ ( 2 ) + cos ϕ s ( 2 ) ] R 2 cos 2 ϕ s ( 2 ) .
r 2 p = r 23 r 2 c r 2 p = r 23 r 2 c .
E d S ( 1 ) d S ( 3 ) = σ ρ 1 cos ϕ ( 1 ) cos ϕ ( 2 ) cos ϕ ( 3 ) cos ϕ s ( 1 ) cos ϕ s ( 2 ) × ( r 1 c r 1 c r 1 p r 1 p ) ( r 2 c r 2 c r 2 p r 2 p )
r 1 c r 1 p r 1 c r 1 p r 2 c r 2 p r 2 c r 2 p = r 1 c r 1 c ( r 2 c r 1 p ) ( r 2 c r 1 p ) 1 r 2 p 1 r 2 p
r 2 c r 1 p = R 2 cos 2 ϕ s ( 2 ) γ ( 2 ) R 2 cos 2 ϕ ( 2 ) + Ω ( 2 ) r 1 p ,
r 2 c r 1 p = R 2 γ ( 2 ) R 2 + Ω ( 2 ) r 1 p
Ω ( 2 ) = γ ( 2 ) cos ϕ ( 2 ) + cos ϕ s ( 2 ) , r 1 p = r 12 r 1 c and r 1 p = r 12 r 1 c .
r c ( 2 ) r p ( 1 ) = cos 2 ϕ s ( 2 ) γ ( 2 ) cos 2 ϕ ( 2 ) ,
r c ( 2 ) r p ( 1 ) = 1 γ ( 2 ) ,
x s = I + z s K x Φ = s J .
κ ( 1 ) κ Φ ( 1 ) κ ( 1 ) κ s ( 1 ) .
τ ( 1 ) = 0.
1 r 1 c = γ ( 1 ) ( r 01 ) + Ω ( 1 ) κ Φ ( 1 ) ; 1 r 1 c = γ ( 1 ) cos 2 ϕ ( 1 ) ( r 01 ) cos 2 ϕ s ( 1 ) + Ω ( 1 ) κ s ( 1 ) cos 2 ϕ s ( 1 ) .
r 1 p = r 12 r 1 c and r 1 p = r 12 r 1 c .
1 r 2 c = γ ( 2 ) ( r 1 p ) + Ω ( 2 ) κ Φ ( 2 ) ; 1 r 2 c = γ ( 2 ) cos 2 ϕ ( 2 ) ( r 1 p ) cos 2 ϕ s ( 2 ) + Ω ( 2 ) κ s ( 2 ) cos 2 ϕ s ( 2 ) .
r 2 p = r 23 r 2 c and r 2 p = r 23 r 2 c .
κ n = κ 1 cos 2 α + κ 2 sin 2 α ,
P ˆ ( 1 ) = A ˆ ( 0 ) × N ˆ ( 1 ) sin ϕ ( 1 )
cos α ( 1 ) = P ˆ ( 1 ) · x Φ ( 1 ) g Φ Φ 1 / 2 ( 1 )
κ ( 1 ) = κ Φ ( 1 ) cos 2 α ( 1 ) + κ s ( 1 ) sin 2 α ( 1 ) , κ ( 1 ) = κ Φ ( 1 ) sin 2 α ( 1 ) + κ s ( 1 ) cos 2 α ( 1 ) .
τ ( 1 ) = [ κ ( 1 ) κ ( 1 ) κ Φ ( 1 ) κ s ( 1 ) ] 1 / 2 .
κ , r w f ( 1 ) = γ ( 1 ) ( r 01 ) + Ω ( 1 ) κ ( 1 ) ;
K , r w f ( 1 ) = γ ( 1 ) cos 2 ϕ ( 1 ) ( r 01 ) cos 2 ϕ s ( 1 ) + Ω ( 1 ) κ ( 1 ) cos 2 ϕ s ( 1 ) ,
τ r w f ( 1 ) = Ω ( 1 ) τ ( 1 ) cos ϕ s ( 1 ) .
κ Φ , r w f ( 1 ) = 1 2 [ κ , r w f ( 1 ) + κ , r w f ( 1 ) ] + 1 2 { [ κ , r w f ( 1 ) κ , r w f ( 1 ) ] 1 + τ r w f 2 ( 1 ) } 1 / 2 ,
κ s , r w f ( 1 ) = 1 2 [ κ , r w f ( 1 ) + κ , r w f ( 1 ) ] 1 2 { [ κ , r w f ( 1 ) κ , r w f ( 1 ) ] 2 + τ r w f 2 ( 1 ) } 1 / 2 .
cos [ 2 α r w f ( 1 ) ] = κ , r w f ( 1 ) κ , r w f ( 1 ) κ Φ , r w f ( 1 ) κ s , r w f ( 1 ) .
( r 1 p ) 1 κ Φ , inc ( 2 ) = 1 κ Φ , r w f ( 1 ) r ( 12 ) , ( r 1 p ) 1 κ s , inc ( 2 ) = 1 K s , r w f ( 1 ) r ( 12 ) .
x Φ , r w f = P ˆ ( 1 ) cos α r w f ( 1 ) + [ A ˆ ( 1 ) × P ˆ ( 1 ) ] sin α r w f ( 1 ) .
K , inc ( 2 ) = κ Φ , inc ( 2 ) cos 2 α inc ( 2 ) + κ s , inc ( 2 ) sin 2 α inc ( 2 ) ,
κ , inc ( 2 ) = κ Φ , inc ( 2 ) sin 2 α inc ( 2 ) + κ s , inc ( 2 ) cos 2 α inc ( 2 ) ,
τ inc ( 2 ) = [ κ , inc ( 2 ) κ , inc ( 2 ) κ Φ , inc ( 2 ) κ s , inc ( 2 ) ] 1 / 2 ,
cos α inc ( 2 ) = x Φ , r w f · A ˆ ( 1 ) × N ˆ 2 sin ϕ ( 2 ) .
κ , r w f ( 2 ) = γ ( 2 ) κ , inc ( 2 ) + Ω ( 2 ) κ ( 2 ) ,
K , r w f ( 2 ) = γ ( 2 ) κ , inc ( 2 ) cos 2 ϕ ( 2 ) cos 2 ϕ s ( 2 ) + Ω ( 2 ) κ ( 2 ) cos 2 ϕ s ( 2 ) ,
τ r w f ( 2 ) = γ ( 2 ) τ inc ( 2 ) cos ϕ ( 2 ) cos ϕ s ( 2 ) + Ω ( 2 ) τ ( 2 ) cos ϕ s ( 2 ) .
r 2 c = 1 κ Φ , r w f and r 2 c = 1 κ s , r w f .
x = I ˆ s cos Φ + J ˆ s sin Φ + K ˆ z ( s ) ,
z ( s ) = c s 2 1 + 1 c 2 e s 2 + j = 0 n A ( j ) s 2 j ,
x s x s = I ˆ cos Φ + J ˆ sin Φ + K ˆ z s , x Φ x Φ = I ˆ s sin Φ + J ˆ s cos Φ ,
z s = s c 1 c 2 e s 2 + j = 0 n ( 2 j ) A ( j ) s 2 j 1 .
g s s x s · x s = 1 + ( z / s ) 2 ,
g Φ Φ x Φ · x Φ = s 2 ,
g s Φ x s · x Φ = 0 ,
g g s s g Φ Φ ( g s Φ ) 2 = s 2 [ 1 + ( z / s ) 2 ] .
N ˆ x s × x Φ / g 1 / 2 = I ˆ cos Φ ( z s ) J ˆ sin Φ ( z s ) + K ˆ [ 1 + ( z / s ) 2 ] 1 / 2 .
x s s 2 x s 2 = K 2 z s 2 , x Φ Φ 2 x Φ 2 = I s cos Φ J s sin Φ , x s Φ x s = I s cos Φ J s sin Φ ,
2 z s 2 = c ( 1 c 2 e s 2 ) 3 / 2 + j = 0 n ( 2 j ) ( 2 j 1 ) A ( j ) s 2 ( j 1 ) .
b s s x s s · N ˆ = ( 2 z / s 2 ) [ 1 + ( z / s ) ] 1 / 2 ,
b Φ Φ x Φ Φ · N = s ( z / s ) [ 1 + ( z / s ) 2 ] 1 / 2 ,
b s Φ x s Φ · N = 0.
g s Φ = 0 and b s Φ = 0 ,
κ s = β s s g s s and κ Φ = b Φ Φ g Φ Φ .
z s = s c 1 c 2 s 2 , g s s = 1 ( 1 c 2 s 2 ) ; g Φ Φ = s 2 , 2 z s 2 = c ( 1 c 2 s 2 ) 3 / 2 , b s s = c ( 1 c 2 s 2 ) ; b Φ Φ = s 2 c ( 1 c 2 s 2 ) .
κ s = c = κ Φ
κ = b 11 g 11 , κ = b 22 g 22 , τ = b 12 ( g 11 g 22 ) 1 / 2 .
g Φ s = 0 and b Φ s = 0 ,
κ Φ = b Φ Φ g Φ Φ , κ s = b s s g s s and τ princ = 0
K = b 12 b 12 b 12 2 g 11 g 22 g 12 2 .
K = κ Φ κ s = b 11 b 22 b 12 2 g 11 g 22 .
K = κ Φ κ s = κ κ τ 2
τ 2 = κ κ κ Φ κ s .
H = 1 2 ( κ Φ + κ s ) = ( g 11 b 22 2 g 12 b 12 + g 22 b 11 ) ( g 11 g 22 g 12 2 ) .
H = g 11 b 22 + g 22 b 11 g 11 g 22 ,
H = 1 2 ( κ Φ + κ s ) = 1 2 ( κ + κ ) .
κ 2 2 H κ + K = 0 ,
κ Φ = 1 2 ( κ + κ ) + 1 2 [ ( κ κ ) 2 + τ 2 ] 1 / 2 , κ s = 1 2 ( κ + κ ) + 1 2 [ ( κ κ ) 2 + τ 2 ] 1 / 2 ,
κ = κ Φ cos 2 α + κ s sin 2 α , κ = κ Φ sin 2 α + κ s cos 2 α .
cos ( 2 α ) = κ κ κ Φ κ s .
E d S 1 d S 2 = σ ( 0 ) ρ 1 cos ϕ ( 1 ) cos ϕ ( 2 ) cos ϕ s ( 1 ) r ( 1 c ) r ( 1 c ) r ( 1 p ) r ( 1 p ) .
E d S 1 d S 2 = σ ρ 1 cos ϕ ( 1 ) cos ϕ ( 2 ) cos ϕ s ( 1 ) { 1 r ( 12 ) [ κ ( 1 c ) + κ ( 1 c ) ] + r 2 ( 12 ) κ ( 1 c ) κ ( 1 c ) } .
E d S 1 d S 2 = σ ρ 1 cos ϕ ( 1 ) cos ϕ ( 2 ) cos ϕ s ( 1 ) { 1 r 12 [ κ ( s ) + κ ( s ) ] + r 12 2 [ κ ( s ) κ ( s ) τ 2 ( s ) ] } .
κ ( s ) = γ r ( o ) + Ω κ ,
κ ( s ) = γ cos 2 ϕ ( 1 ) r ( o ) cos 2 ϕ s ( 1 ) + Ω κ cos 2 ϕ s ( 1 ) ,
τ ( 2 ) = τ ( o ) γ cos ϕ ( 1 ) cos ϕ s ( 1 ) + Ω τ cos ϕ s ( 1 ) ,
[ κ ( s ) + κ ( s ) ] cos ϕ s ( 1 ) = γ [ cos 2 ϕ ( 1 ) + cos 2 ϕ s ( 1 ) ] r ( o ) cos ϕ s ( 1 ) + Ω κ [ cos 2 ϕ s ( 1 ) + κ ] cos ϕ s ( 1 ) .
[ κ ( s ) + κ ( s ) ] cos ϕ s ( 1 ) = { γ [ cos 2 ϕ ( 1 ) + cos 2 ϕ s ( 1 ) ] r ( o ) cos ϕ s ( 1 ) + [ 2 H cos ϕ s ( 1 ) + γ 2 κ sin 2 ϕ ( 1 ) ] cos ϕ s ( 1 ) } .
[ κ ( s ) κ ( s ) τ 2 ( s ) ] cos ϕ s ( 1 ) = γ 2 cos 2 ϕ ( 1 ) r 2 ( o ) cos ϕ s ( 1 ) + γ Ω [ κ + κ cos 2 ϕ ( 1 ) ] r ( o ) cos ϕ s ( 1 ) + Ω 2 ( κ κ τ 2 ) cos ϕ s ( 1 ) .
κ ( 1 c ) κ ( 1 c ) cos ϕ s ( 1 ) = { γ 2 cos 2 ϕ ( 1 ) r 2 ( o ) cos ϕ s ( 1 ) + γ Ω [ 2 H cos 2 ϕ s ( 1 ) + γ 2 κ sin 2 ϕ ( 1 ) ] r ( o ) cos ϕ s ( 1 ) + Ω 2 K cos ϕ s ( 1 ) } ,
E d S 1 d S 2 = σ ρ cos ϕ ( 1 ) cos ϕ ( 2 ) cos ϕ ( 1 ) + r 12 L 1 ( 1 ) + r 12 2 L 2 ( 1 ) ,
L 1 ( 1 ) = 2 ( 2 H cos 2 ϕ ( 1 ) + K N sin 2 ϕ ) ( 1 ) + 2 cos ϕ ( 1 ) / r ( o ) , L 2 ( 1 ) = 4 K cos ϕ 2 ( 2 H cos 2 ϕ + K N sin 2 ϕ ) / r ( o ) [ cos ϕ ( 1 ) ] / r 2 ( o ) .
K N sin 2 ϕ ( 1 ) = i , j = 1 2 a i a j b i j ,
b i j = ( x u × x υ · 2 x u υ ) / g 1 / 2 , a j = g j 1 A ˆ ( o ) · x u + g j 2 A ˆ ( o ) · x υ , g 11 = g 22 / g , g 22 = g 11 / g , g 12 = g 12 / g .

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