Abstract

A ray-theoretic account of the passage of light through a radially inhomogeneous transparent sphere has been used to establish the existence of multiple primary rainbows for some refractive index profiles. The existence of such additional bows is a consequence of a sufficiently attractive potential in the interior of the drop, i.e., the refractive index gradient should be sufficiently negative there. The profiles for which this gradient is monotonically increasing do not result in this phenomenon, but nonmonotone profiles can do so, depending on the form of n. Sufficiently oscillatory profiles can lead to apparently singular behavior in the deviation angle (within the geometrical optics approximation) as well as multiple rainbows. These results also apply to systems with circular cylindrical cross sections, and may be of value in the field of rainbow refractometry.

© 2007 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. C. L. Adler, J. A. Lock, I. P. Rafferty, and W. Hickok, "Twin-rainbow metrology. I. Measurement of the thickness of a thin liquid film draining under gravity," Appl. Opt. 42, 6584-6594 (2003).
    [CrossRef] [PubMed]
  2. C. W. Chan and W. K. Lee, "Measurement of a liquid refractive index by using high-order rainbows," J. Opt. Soc. Am. B 13, 532-535 (1996).
    [CrossRef]
  3. H. Hattori, H. Yamanaka, H. Kurniawan, S. Yokoi, and K. Kagawa, "Using minimum deviation of a secondary rainbow and its application to water analysis in a high-precision refractive-index comparator for liquids," Appl. Opt. 36, 5552-5556 (1997).
    [CrossRef] [PubMed]
  4. H. Hattori, K. Kakui, H. Kurniawan, and K. Kagawa, "Liquid refractometry by the rainbow method," Appl. Opt. 37, 4123-4129 (1998).
    [CrossRef]
  5. H. Hattori, "Simulation study on refractometry by the rainbow method," Appl. Opt. 38, 4037-4046 (1999).
    [CrossRef]
  6. J. Hom and N. Chigier, "Rainbow refractometry: simultaneous measurement of temperature, refractive index, and size of droplets," Appl. Opt. 41, 1899-1907 (2002).
    [CrossRef] [PubMed]
  7. C. L. Adler, J. A. Lock, and B. R. Stone, "Rainbow scattering by a cylinder with a nearly elliptical cross section," Appl. Opt. 37, 1540-1550 (1998).
    [CrossRef]
  8. C. L. Adler, J. A. Lock, D. Phipps, K. Saunders, and J. Nash, "Supernumerary spacings of rainbows produced by an elliptical cross-section cylinder. II: Experiment," Appl. Opt. 40, 2535-2545 (2001).
    [CrossRef]
  9. P. Massoli, "Rainbow refractometry applied to radially inhomogeneous spheres: the critical case of evaporating droplets," Appl. Opt. 37, 3227-3235 (1998).
    [CrossRef]
  10. M. Schneider and E. D. Hirleman, "Influence of internal refractive index gradients on size measurements of spherically symmetric particles by phase Doppler anemometry," Appl. Opt. 33, 2379-2388 (1994).
    [CrossRef] [PubMed]
  11. M. Schneider, E. D. Hirleman, H. Salaheen, D. Q. Choudury, and S. C. Hill, "Rainbows and radially inhomogeneous droplets," in Proceedings of the Third International Congress on Optical Particle Sizing, M. Maeda, ed. (Yokohama, 1993), pp. 323-326.
  12. P. Massoli, "Temperature and size of droplets inferred by light scattering methods: a theoretical analysis of the influence of internal inhomogeneities," presented at the 13th Annual Conference on Liquid Atomization and Spray Systems (ILASS- Europe, Florence, 1997).
  13. L. Kai, P. Massoli, and A. D'Alessio, "Some far-field scattering characteristics of radially inhomogeneous particles," Part. Part. Syst. Charact. 11, 385-390 (1994).
    [CrossRef]
  14. P. L. Marston, "Rainbow phenomena and the detection of nonsphericity in drops," Appl. Opt. 19, 680-685 (1980).
    [CrossRef] [PubMed]
  15. J. P. A. J. van Beeck and M. L. Riethmuller, "Rainbow phenomena applied to the measurement of droplet size and velocity and to the detection of nonsphericity," Appl. Opt. 35, 2259-2266 (1996).
    [CrossRef] [PubMed]
  16. J. P. A. J. van Beeck and M. L. Riethmuller, "Nonintrusive measurements of temperature and size of single falling raindrops," Appl. Opt. 34, 1633-1639 (1995).
    [CrossRef] [PubMed]
  17. K. Anders, N. Roth, and A. Frohn, "Influence of refractive index gradients within droplets on rainbow position and implications for rainbow refractometry," Part. Part. Syst. Charact. 13, 125-129 (1996).
    [CrossRef]
  18. N. Roth, K. Anders, and A. Frohn, "Size insensitive rainbow refractometry: theoretical aspects," presented at the Eighth International Symposium on Applications of Laser Techniques to Fluid Mechanics (Lisbon, 1996).
  19. J. W. Y. Lit, "Radius of uncladded optical fiber from back-scattered radiation pattern," J. Opt. Soc. Am. 65, 1311-1315 (1975).
    [CrossRef]
  20. D. Marcuse, "Light scattered from unclad fibers: ray theory," Appl. Opt. 14, 1528-1532 (1975).
    [CrossRef] [PubMed]
  21. H. M. Presby, "Refractive index and diameter measurements of unclad optical fibers," J. Opt. Soc. Am. 64, 280-284 (1974).
    [CrossRef]
  22. M. R. Vetrano, J. P. A. J. van Beeck, and M. L. Riethmuller, "Generalization of the rainbow Airy theory to nonuniform spheres," Opt. Lett. 30, 658-660 (2005).
    [CrossRef] [PubMed]
  23. M. R. Vetrano, J. P. A. J. van Beeck, and M. L. Riethmuller, "Assessment of refractive index gradients by standard rainbow thermometry," Appl. Opt. 44, 7275-7281 (2005).
    [CrossRef] [PubMed]
  24. M. Born and E. Wolf, Principles of Optics (Cambridge U. Press, 2002).
  25. R. K. Luneberg, Mathematical Theory of Optics (U. California Press, 1966).
  26. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products (Academic, 1980).
  27. C. L. Brockman and N. G. Alexopoulos, "Geometrical optics of inhomogeneous particles: glory ray and the rainbow revisited," Appl. Opt. 16, 166-174 (1977).
    [CrossRef] [PubMed]

2005 (2)

2003 (1)

2002 (1)

2001 (1)

1999 (1)

1998 (3)

1997 (1)

1996 (3)

1995 (1)

1994 (2)

L. Kai, P. Massoli, and A. D'Alessio, "Some far-field scattering characteristics of radially inhomogeneous particles," Part. Part. Syst. Charact. 11, 385-390 (1994).
[CrossRef]

M. Schneider and E. D. Hirleman, "Influence of internal refractive index gradients on size measurements of spherically symmetric particles by phase Doppler anemometry," Appl. Opt. 33, 2379-2388 (1994).
[CrossRef] [PubMed]

1980 (1)

1977 (1)

1975 (2)

1974 (1)

Adler, C. L.

Alexopoulos, N. G.

Anders, K.

K. Anders, N. Roth, and A. Frohn, "Influence of refractive index gradients within droplets on rainbow position and implications for rainbow refractometry," Part. Part. Syst. Charact. 13, 125-129 (1996).
[CrossRef]

N. Roth, K. Anders, and A. Frohn, "Size insensitive rainbow refractometry: theoretical aspects," presented at the Eighth International Symposium on Applications of Laser Techniques to Fluid Mechanics (Lisbon, 1996).

Born, M.

M. Born and E. Wolf, Principles of Optics (Cambridge U. Press, 2002).

Brockman, C. L.

Chan, C. W.

Chigier, N.

Choudury, D. Q.

M. Schneider, E. D. Hirleman, H. Salaheen, D. Q. Choudury, and S. C. Hill, "Rainbows and radially inhomogeneous droplets," in Proceedings of the Third International Congress on Optical Particle Sizing, M. Maeda, ed. (Yokohama, 1993), pp. 323-326.

D'Alessio, A.

L. Kai, P. Massoli, and A. D'Alessio, "Some far-field scattering characteristics of radially inhomogeneous particles," Part. Part. Syst. Charact. 11, 385-390 (1994).
[CrossRef]

Frohn, A.

K. Anders, N. Roth, and A. Frohn, "Influence of refractive index gradients within droplets on rainbow position and implications for rainbow refractometry," Part. Part. Syst. Charact. 13, 125-129 (1996).
[CrossRef]

N. Roth, K. Anders, and A. Frohn, "Size insensitive rainbow refractometry: theoretical aspects," presented at the Eighth International Symposium on Applications of Laser Techniques to Fluid Mechanics (Lisbon, 1996).

Gradshteyn, I. S.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products (Academic, 1980).

Hattori, H.

Hickok, W.

Hill, S. C.

M. Schneider, E. D. Hirleman, H. Salaheen, D. Q. Choudury, and S. C. Hill, "Rainbows and radially inhomogeneous droplets," in Proceedings of the Third International Congress on Optical Particle Sizing, M. Maeda, ed. (Yokohama, 1993), pp. 323-326.

Hirleman, E. D.

M. Schneider and E. D. Hirleman, "Influence of internal refractive index gradients on size measurements of spherically symmetric particles by phase Doppler anemometry," Appl. Opt. 33, 2379-2388 (1994).
[CrossRef] [PubMed]

M. Schneider, E. D. Hirleman, H. Salaheen, D. Q. Choudury, and S. C. Hill, "Rainbows and radially inhomogeneous droplets," in Proceedings of the Third International Congress on Optical Particle Sizing, M. Maeda, ed. (Yokohama, 1993), pp. 323-326.

Hom, J.

Kagawa, K.

Kai, L.

L. Kai, P. Massoli, and A. D'Alessio, "Some far-field scattering characteristics of radially inhomogeneous particles," Part. Part. Syst. Charact. 11, 385-390 (1994).
[CrossRef]

Kakui, K.

Kurniawan, H.

Lee, W. K.

Lit, J. W. Y.

Lock, J. A.

Luneberg, R. K.

R. K. Luneberg, Mathematical Theory of Optics (U. California Press, 1966).

Marcuse, D.

Marston, P. L.

Massoli, P.

P. Massoli, "Rainbow refractometry applied to radially inhomogeneous spheres: the critical case of evaporating droplets," Appl. Opt. 37, 3227-3235 (1998).
[CrossRef]

L. Kai, P. Massoli, and A. D'Alessio, "Some far-field scattering characteristics of radially inhomogeneous particles," Part. Part. Syst. Charact. 11, 385-390 (1994).
[CrossRef]

P. Massoli, "Temperature and size of droplets inferred by light scattering methods: a theoretical analysis of the influence of internal inhomogeneities," presented at the 13th Annual Conference on Liquid Atomization and Spray Systems (ILASS- Europe, Florence, 1997).

Nash, J.

Phipps, D.

Presby, H. M.

Rafferty, I. P.

Riethmuller, M. L.

Roth, N.

K. Anders, N. Roth, and A. Frohn, "Influence of refractive index gradients within droplets on rainbow position and implications for rainbow refractometry," Part. Part. Syst. Charact. 13, 125-129 (1996).
[CrossRef]

N. Roth, K. Anders, and A. Frohn, "Size insensitive rainbow refractometry: theoretical aspects," presented at the Eighth International Symposium on Applications of Laser Techniques to Fluid Mechanics (Lisbon, 1996).

Ryzhik, I. M.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products (Academic, 1980).

Salaheen, H.

M. Schneider, E. D. Hirleman, H. Salaheen, D. Q. Choudury, and S. C. Hill, "Rainbows and radially inhomogeneous droplets," in Proceedings of the Third International Congress on Optical Particle Sizing, M. Maeda, ed. (Yokohama, 1993), pp. 323-326.

Saunders, K.

Schneider, M.

M. Schneider and E. D. Hirleman, "Influence of internal refractive index gradients on size measurements of spherically symmetric particles by phase Doppler anemometry," Appl. Opt. 33, 2379-2388 (1994).
[CrossRef] [PubMed]

M. Schneider, E. D. Hirleman, H. Salaheen, D. Q. Choudury, and S. C. Hill, "Rainbows and radially inhomogeneous droplets," in Proceedings of the Third International Congress on Optical Particle Sizing, M. Maeda, ed. (Yokohama, 1993), pp. 323-326.

Stone, B. R.

van Beeck, J. P. A. J.

Vetrano, M. R.

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Cambridge U. Press, 2002).

Yamanaka, H.

Yokoi, S.

Appl. Opt. (15)

D. Marcuse, "Light scattered from unclad fibers: ray theory," Appl. Opt. 14, 1528-1532 (1975).
[CrossRef] [PubMed]

C. L. Brockman and N. G. Alexopoulos, "Geometrical optics of inhomogeneous particles: glory ray and the rainbow revisited," Appl. Opt. 16, 166-174 (1977).
[CrossRef] [PubMed]

P. L. Marston, "Rainbow phenomena and the detection of nonsphericity in drops," Appl. Opt. 19, 680-685 (1980).
[CrossRef] [PubMed]

M. Schneider and E. D. Hirleman, "Influence of internal refractive index gradients on size measurements of spherically symmetric particles by phase Doppler anemometry," Appl. Opt. 33, 2379-2388 (1994).
[CrossRef] [PubMed]

P. Massoli, "Rainbow refractometry applied to radially inhomogeneous spheres: the critical case of evaporating droplets," Appl. Opt. 37, 3227-3235 (1998).
[CrossRef]

H. Hattori, "Simulation study on refractometry by the rainbow method," Appl. Opt. 38, 4037-4046 (1999).
[CrossRef]

J. P. A. J. van Beeck and M. L. Riethmuller, "Nonintrusive measurements of temperature and size of single falling raindrops," Appl. Opt. 34, 1633-1639 (1995).
[CrossRef] [PubMed]

J. P. A. J. van Beeck and M. L. Riethmuller, "Rainbow phenomena applied to the measurement of droplet size and velocity and to the detection of nonsphericity," Appl. Opt. 35, 2259-2266 (1996).
[CrossRef] [PubMed]

C. L. Adler, J. A. Lock, and B. R. Stone, "Rainbow scattering by a cylinder with a nearly elliptical cross section," Appl. Opt. 37, 1540-1550 (1998).
[CrossRef]

H. Hattori, H. Yamanaka, H. Kurniawan, S. Yokoi, and K. Kagawa, "Using minimum deviation of a secondary rainbow and its application to water analysis in a high-precision refractive-index comparator for liquids," Appl. Opt. 36, 5552-5556 (1997).
[CrossRef] [PubMed]

H. Hattori, K. Kakui, H. Kurniawan, and K. Kagawa, "Liquid refractometry by the rainbow method," Appl. Opt. 37, 4123-4129 (1998).
[CrossRef]

C. L. Adler, J. A. Lock, D. Phipps, K. Saunders, and J. Nash, "Supernumerary spacings of rainbows produced by an elliptical cross-section cylinder. II: Experiment," Appl. Opt. 40, 2535-2545 (2001).
[CrossRef]

J. Hom and N. Chigier, "Rainbow refractometry: simultaneous measurement of temperature, refractive index, and size of droplets," Appl. Opt. 41, 1899-1907 (2002).
[CrossRef] [PubMed]

C. L. Adler, J. A. Lock, I. P. Rafferty, and W. Hickok, "Twin-rainbow metrology. I. Measurement of the thickness of a thin liquid film draining under gravity," Appl. Opt. 42, 6584-6594 (2003).
[CrossRef] [PubMed]

M. R. Vetrano, J. P. A. J. van Beeck, and M. L. Riethmuller, "Assessment of refractive index gradients by standard rainbow thermometry," Appl. Opt. 44, 7275-7281 (2005).
[CrossRef] [PubMed]

J. Opt. Soc. Am. (2)

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

Opt. Lett. (1)

Part. Part. Syst. Charact. (2)

L. Kai, P. Massoli, and A. D'Alessio, "Some far-field scattering characteristics of radially inhomogeneous particles," Part. Part. Syst. Charact. 11, 385-390 (1994).
[CrossRef]

K. Anders, N. Roth, and A. Frohn, "Influence of refractive index gradients within droplets on rainbow position and implications for rainbow refractometry," Part. Part. Syst. Charact. 13, 125-129 (1996).
[CrossRef]

Other (6)

N. Roth, K. Anders, and A. Frohn, "Size insensitive rainbow refractometry: theoretical aspects," presented at the Eighth International Symposium on Applications of Laser Techniques to Fluid Mechanics (Lisbon, 1996).

M. Born and E. Wolf, Principles of Optics (Cambridge U. Press, 2002).

R. K. Luneberg, Mathematical Theory of Optics (U. California Press, 1966).

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products (Academic, 1980).

M. Schneider, E. D. Hirleman, H. Salaheen, D. Q. Choudury, and S. C. Hill, "Rainbows and radially inhomogeneous droplets," in Proceedings of the Third International Congress on Optical Particle Sizing, M. Maeda, ed. (Yokohama, 1993), pp. 323-326.

P. Massoli, "Temperature and size of droplets inferred by light scattering methods: a theoretical analysis of the influence of internal inhomogeneities," presented at the 13th Annual Conference on Liquid Atomization and Spray Systems (ILASS- Europe, Florence, 1997).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (8)

Fig. 1
Fig. 1

Ray path for a single internal reflection; any point on the path is identified by its polar coordinates ( ξ , θ ) .

Fig. 2
Fig. 2

F(i) defined by Eq. (18a) is the incident angular component of the additional deviation incurred for the inhomogeneous sphere over that for the homogeneous one [see Eq. (18)], and is plotted for n 0 = 5 / 3 . F1(i) is for n 0 = 2.5 .

Fig. 3
Fig. 3

Graphs of (i) the exact ray deviation TotD(i) found from Eq. (10) for the profile n ¯ ( λ ) = ( a λ + b ) 1 with a = ε ( = 0.25   here ) and n 0 = 5 / 3 , corresponding to n 1 = 5 / ( 5 ε + 3 ) ; (ii) the deviation for the homogeneous sphere Dh(i) for n ¯ ( λ ) = n 0 = 5 / 3 ; (iii) the additional deviation ε F ( i ) , due to the inhomogeneity [see Eqs. (18) and (18a)]; (iv) the linear approximation to the deviation Dh ( i ) + ε F ( i ) , as calculated from Eq. (18).

Fig. 4
Fig. 4

Graphs of symmetric refractive index profiles.

Fig. 5
Fig. 5

Graphs of D(i) for two symmetric refractive index profiles ρ 1 ( λ ) and ρ 2 ( λ ) .

Fig. 6
Fig. 6

(a) Variation of refractive index n ( λ ) as a function of the normalized radius λ of the sphere. (b) Variation of singular point ρ ( λ ) as a function of the normalized radius λ of the sphere.

Fig. 7
Fig. 7

Graph of D(i) for n ¯ ( λ ) = [ 5 + sin ( 6 π λ ) ] / 3 . Note the multiple extrema and also the apparent singular behavior near i arcsin 0.76 = 49.5 ° , corresponding to the smallest minimum of ρ(λ) in Fig. 6.

Fig. 8
Fig. 8

Graphs of (i) D(i) as given by Eq. (10), (ii) the integral term L(i) as defined by Eq. (24), and (iii) the linear part of D(i), namely D l ( i ) = 2 i , where D ( i ) = D l ( i ) + 4 L ( i ) π .

Equations (42)

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

η × n ( η ) s = constant .
n ( η ) η   sin   ϕ = constant = K ,
sin   ϕ = η ( θ ) η 2 ( θ ) + ( d η / d θ ) 2 .
d η d θ = ± η K η 2 n 2 ( η ) K 2 ,
θ θ 0 = ± K R ξ d η η η 2 n 2 ( η ) K 2 .
D ( i ) = 2 i + π 4 r ( i ) + Θ ,
d ξ d θ | θ ¯ = 0.
d θ d ξ = K ξ ξ 2 n 2 ( ξ ) K 2 .
K ¯ ( i ) = lim λ 1 [ λ n ( λ ) ] sin   r ( i ) = lim λ 1 λ   sin   i = sin   i , but   when   λ = λ ¯ ,
K ¯ = λ ¯ n ( λ ¯ ) sin [ ϕ ( λ ¯ ) ] = λ ¯ n ( λ ¯ ) sin [ π / 2 ] = λ ¯ n ( λ ¯ ) ,
K ¯ = λ ¯ n ( λ ¯ ) = sin   i .
θ ¯ i = K ¯ 1 λ ¯ d η η η 2 n 2 ( η ) K 2 = K ¯ λ ¯ 1 d λ λ λ 2 n ¯ 2 ( λ ) K ¯ 2 K ¯ I ( λ ¯ , i ) ,
β = π + 2 ( r + θ ¯ i ) ,
D ( i ) = 2 i π + 4 K ¯ λ ¯ 1 d λ λ λ 2 n ¯ 2 ( λ ) K ¯ 2 .
β = π 2 ( r + θ ¯ i ) .
n ¯ ( λ ) = ( a λ + b ) 1 n 0 n 1 ( n 0 n 1 ) λ + n 1 .
λ ¯ = n 1   sin   i n 0 n 1 ( n 0 n 1 ) sin   i .
I ( λ ¯ , i ) = λ ¯ 1 ( a λ + b ) d λ λ λ 2 K ¯ 2 ( a λ + b ) 2 a I A + b I B ,
w h e r e I A = λ ¯ 1 d λ C λ 2 + B λ + A ,
I B = λ ¯ 1 d λ λ C λ 2 + B λ + A ,
C 1 a 2 K ¯ 2 = ( 1 [ n 0 n 1 n 0 n 1 ] 2 sin 2 i ) > 0 ,
B 2 a b K ¯ 2 = 2 [ n 0 n 1 n 0 n 1 ] sin 2 i ,
A b 2 K ¯ 2 .
I ( λ , i ) = a   ln [ 2 C ( C λ 2 + B λ + A ) + 2 C λ + B ] C + b A  arcsin   2 A + B λ λ B 2 4 A C .
I ( λ ¯ , i ) = λ ¯ 1 ( a + b λ ) [ 1 λ 2 K ¯ 2 b 2 + 2 ε λ b K ¯ 2 ( λ 2 K ¯ 2 b 2 ) 3 / 2 ] d λ I 1 + I 2 + I 3 + I 4 ,
I 1 = a d λ λ 2 K ¯ 2 b 2 = a   ln ( λ + λ 2 K ¯ 2 b 2 ) ;
I 2 = b d λ λ λ 2 K ¯ 2 b 2 = 1 K ¯  arcsin ( K ¯ b λ ) ;
I 3 = 2 ε a b K ¯ 2 λ d λ ( λ 2 K ¯ 2 b 2 ) 3 / 2 = 2 ε a b K ¯ 2 λ 2 K ¯ 2 b 2 ;
I 4 = 2 ε b 2 K ¯ 2 d λ ( λ 2 K ¯ 2 b 2 ) 3 / 2 = 2 ε λ λ 2 K ¯ 2 b 2 .
λ ¯ = ( ε λ ¯ + b ) sin   i , i .e . , λ ¯ = b   sin   i ( 1 + ε   sin   i ) + O ( ε 2 ) .
[ I 1 ] λ ¯ 1 = ε  ln ( 1 + ( λ 2 K ¯ 2 b 2 ) 1 / 2 b   sin   i ) + O ( ε 3 / 2 ) ;
[ I 2 ] λ ¯ 1 = 1 K ¯ [ arcsin ( K ¯ b ) π 2 + ( 2 ε   sin   i ) 1 / 2 ] + O ( ε 3 / 2 ) ;
[ I 3 ] λ ¯ 1 = O ( ε 3 / 2 ) ;
[ I 4 ] λ ¯ 1 = ( 2 ε sin   i ) 1 / 2 2 ε ( 1 K ¯ 2 b 2 ) 1 / 2 + O ( ε 3 / 2 ) .
D ( i ) = 2 i π + 4 K ¯ { ε   ln ( 1+ ( 1 - K ¯ 2 b 2 ) 1 / 2 b  sin   i ) 1 K ¯ [ arcsin ( K ¯ b ) π 2 + ( 2 ε   sin   i ) 1 / 2 ] + ( 2 ε sin   i ) 1 / 2 2 ε ( 1 K ¯ 2 b 2 ) 1 / 2 } + O ( ε 3 / 2 )
2 i + π 4   arcsin ( K ¯ b ) + 4 ε K ¯ × { ln ( 1 + ( 1 K ¯ 2 b 2 ) 1 / 2 b   sin   i ) 2 ( 1 K ¯ 2 b 2 ) 1 / 2 }
= 2 i + π 4   arcsin ( b   sin   i ) + 4 ε   sin   i × { ln ( 1 + ( 1 b 2 sin 2 i ) 1 / 2 b   sin   i ) 2 ( 1 b 2 sin 2 i ) 1 / 2 } .
D h ( i ) + ε F ( i ) ,
where   F ( i ) = 4   sin   i { ln ( 1 + ( 1 b 2 sin 2 i ) 1 / 2 b   sin   i ) 2 ( 1 b 2 sin 2 i ) 1 / 2 } ,
δ D h ( i c ) + ε [ F ( i c ) + δ F ( i c ) ] 0 , or
δ ε F ( i c ) D h ( i c ) + ε F ( i c ) = ε F ( i c ) D h ( i c ) + O ( ε 2 ) .
L ( i ) = K ¯ λ ¯ 1 d λ λ λ 2 n ¯ 2 ( λ ) K ¯ 2

Metrics