Abstract

Light scattered by an oblate drop of water has been observed to produce cusp caustics in the general vicinity of the rainbow region [ P. L. Marston and E. H. Trinh, Nature London 312, 529– 531 ( 1984)]. The principal curvatures of the generic local wave front that produces the far-field transverse cusp are examined. This wave front is shown to generate a transverse cusp curve (UUc)3 = − dV2, where U and V are horizontal and vertical scattering angles and Uc is the cusp point direction. The far-field opening rate d is calculated for the transverse cusp. It is shown that d has a simple dependence on the parameters of the generic wave front. We define the aspect ratio of the drop q = D/H, where H is the height and D is the equatorial width for the scattering drop. The method of generalized ray tracing is used to relate q to principal curvatures and shape parameters of the outgoing wavefront and hence to d. Measurements of d. for scattering laser light from acoustically levitated drops appear to support the calculation. As q goes to q4 ≈ 1.31, the critical value for generation of a hyperbolic–umbilic focal section, the predicted d goes to infinity. The nature of the divergence was numerically investigated as was the rate at which d vanishes as q approaches critical values for lips and transition events.

© 1991 Optical Society of America

Full Article  |  PDF Article

Corrections

Cleon E. Dean and Philip L. Marston, "Opening rate of the transverse cusp diffraction catastrophe in light scattered by oblate spheroidal drops: errata," Appl. Opt. 32, 2163-2163 (1993)
https://www.osapublishing.org/ao/abstract.cfm?uri=ao-32-12-2163

References

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  1. P. L. Marston, E. H. Trinh, “Hyperbolic umbilic diffraction catastrophe and rainbow scattering from drops,” Nature (London) 312, 529–531 (1984).
    [CrossRef]
  2. P. L. Marston, “Cusp diffraction catastrophes from spheroids: generalized rainbows and inverse scattering,” Opt. Lett. 10, 588–590 (1985).
    [CrossRef] [PubMed]
  3. P. L. Marston, C. E. Dean, H. J. Simpson, “Light scattering from spheroidal drops: exploring optical catastrophes and generalized rainbows,” AIP Conf. Proc. 197, 275–285 (1989). [Because of printing errors, replace (γ/α) in Eq. (4) by (γ/α)1/2 and κ1 in Eq. (6) by κ2.]
    [CrossRef]
  4. J. F. Nye, “Rainbow scattering from spheroidal drops—an explanation of the hyperbolic umbilic foci,” Nature (London) 312, 531–532 (1984).
    [CrossRef]
  5. P. L. Marston, “Transverse cusp diffraction catastrophes: some pertinent wavefronts and a Pearcey approximation to the wavefield,” J. Acoust. Soc. Am. 81, 226–232 (1987); J. Acoust. Soc. Am. (E) 83, 1976 (1988).
    [CrossRef]
  6. H. J. Simpson, P. L. Marston, “Scattering of white light from levitated oblate water drops near rainbows and other diffraction catastrophes,” Appl. Opt. 30, 3466–3472 (1991).
    [CrossRef]
  7. O. N. Stavroudis, The Optics of Rays, Wavefronts, and Caustics (Academic, New York, 1972), pp. 161–179; R. S. Chang, O. N. Stavroudis, “Generalized ray tracing, caustic surfaces, generalized bending, and the construction of a novel merit function for optical design,” J. Opt. Soc. Am. 70, 976–985 (1980); D. G. Burkhard, D. L. Shealy, “Flux density for ray propagation in geometrical optics,” J. pt. Soc. Am. 63, 299–304 (1973); M. Herzberger, Modern Geometrical Optics (Wiley-Interscience, New York, 1958).
    [CrossRef]
  8. M. V. Berry, “Waves and Thom’s theorem,” Adv. Phys. 25, 1–26 (1976).
    [CrossRef]
  9. M. V. Berry, C. Upstill, “Catastrophe optics: morphologies of caustics and their diffraction patterns,” Prog. Opt. 18, 258–346 (1980).
  10. C. E. Dean, “Analysis of scattered light: II. The opening rate of the transverse cusp from oblate drops,” Ph.D. dissertation (Washington State University, Pullman, Wash., 1989).
  11. W. J. Wiscombe, A. Mugnai, “Scattering from nonspherical Chebyshev particles. 2: Means of angular scattering patterns,” Appl. Opt. 27, 2405–2421 (1988); S. Asano, G. Yamamoto, “Light scattering by a spheroidal particle,” Appl. Opt. 14, 29–49 (1975).
    [CrossRef] [PubMed]
  12. P. L. Marston, “Hyperbolic-umbilic diffraction catastrophes and the tracing of local principal curvatures of wave fronts,” J. Acoust. Soc. Am. Suppl. 80, S73 (1986); “Hyperbolic-umbilic focal sections: the wavefield and the merging of rays at caustic lines,” J. Acoust. Soc. Am. Suppl. 81, S14 (1987).
    [CrossRef]
  13. P. L. Marston, “Wavefront geometries giving transverse cusp and hyperbolic umbilic foci in acoustic shocks,” in Shock Waves in Condensed Matter 1987, S. C. Schmidt, N. C. Holmes, eds. (Elsevier, Amsterdam, 1988), pp. 203–206.
  14. J. F. Nye, J. H. Hannay, “The orientations and distortions of caustics in geometrical optics,” Opt. Acta 31, 115–130 (1984).
    [CrossRef]
  15. J. A. Kneisly, “Local curvature of wavefronts in an optical system,” J. Opt. Soc. Am. 54, 229–235 (1964).
    [CrossRef]
  16. A. Gullstrand, “Die reele optische Abbildung,” Sven. Vetensk. Handl. 41, 1–119 (1906).
  17. J. F. Nye, “Optical caustics in the near field from liquid drops,” Proc. R. Soc. London Ser. A 361, 21–41 (1978).
    [CrossRef]
  18. T. Pearcey, “The structure of an electromagnetic field in the neighborhood of a cusp of a caustic,” Philos. Mag. 37, 311–317 (1946).
  19. H. J. Simpson, “The lips event for light backscattered from levitated water drops,” M.S. degree project report (Washington State University, Pullman, Wash., 1988).
  20. G. Dangelmayr, F. J. Wright, “On the validity of the paraxial eikonal in catastrophe optics,” J. Phys. A 17, 99–108 (1984).
    [CrossRef]
  21. D. S. Langley, P. L. Marston, “Critical-angle scattering of laser light from bubbles in water: measurements, models, and application to sizing of bubbles,” Appl. Opt. 23, 1044–1054 (1984).
    [CrossRef] [PubMed]

1991 (1)

H. J. Simpson, P. L. Marston, “Scattering of white light from levitated oblate water drops near rainbows and other diffraction catastrophes,” Appl. Opt. 30, 3466–3472 (1991).
[CrossRef]

1989 (1)

P. L. Marston, C. E. Dean, H. J. Simpson, “Light scattering from spheroidal drops: exploring optical catastrophes and generalized rainbows,” AIP Conf. Proc. 197, 275–285 (1989). [Because of printing errors, replace (γ/α) in Eq. (4) by (γ/α)1/2 and κ1 in Eq. (6) by κ2.]
[CrossRef]

1988 (1)

1987 (1)

P. L. Marston, “Transverse cusp diffraction catastrophes: some pertinent wavefronts and a Pearcey approximation to the wavefield,” J. Acoust. Soc. Am. 81, 226–232 (1987); J. Acoust. Soc. Am. (E) 83, 1976 (1988).
[CrossRef]

1986 (1)

P. L. Marston, “Hyperbolic-umbilic diffraction catastrophes and the tracing of local principal curvatures of wave fronts,” J. Acoust. Soc. Am. Suppl. 80, S73 (1986); “Hyperbolic-umbilic focal sections: the wavefield and the merging of rays at caustic lines,” J. Acoust. Soc. Am. Suppl. 81, S14 (1987).
[CrossRef]

1985 (1)

1984 (5)

D. S. Langley, P. L. Marston, “Critical-angle scattering of laser light from bubbles in water: measurements, models, and application to sizing of bubbles,” Appl. Opt. 23, 1044–1054 (1984).
[CrossRef] [PubMed]

P. L. Marston, E. H. Trinh, “Hyperbolic umbilic diffraction catastrophe and rainbow scattering from drops,” Nature (London) 312, 529–531 (1984).
[CrossRef]

G. Dangelmayr, F. J. Wright, “On the validity of the paraxial eikonal in catastrophe optics,” J. Phys. A 17, 99–108 (1984).
[CrossRef]

J. F. Nye, J. H. Hannay, “The orientations and distortions of caustics in geometrical optics,” Opt. Acta 31, 115–130 (1984).
[CrossRef]

J. F. Nye, “Rainbow scattering from spheroidal drops—an explanation of the hyperbolic umbilic foci,” Nature (London) 312, 531–532 (1984).
[CrossRef]

1980 (1)

M. V. Berry, C. Upstill, “Catastrophe optics: morphologies of caustics and their diffraction patterns,” Prog. Opt. 18, 258–346 (1980).

1978 (1)

J. F. Nye, “Optical caustics in the near field from liquid drops,” Proc. R. Soc. London Ser. A 361, 21–41 (1978).
[CrossRef]

1976 (1)

M. V. Berry, “Waves and Thom’s theorem,” Adv. Phys. 25, 1–26 (1976).
[CrossRef]

1964 (1)

1946 (1)

T. Pearcey, “The structure of an electromagnetic field in the neighborhood of a cusp of a caustic,” Philos. Mag. 37, 311–317 (1946).

1906 (1)

A. Gullstrand, “Die reele optische Abbildung,” Sven. Vetensk. Handl. 41, 1–119 (1906).

Berry, M. V.

M. V. Berry, C. Upstill, “Catastrophe optics: morphologies of caustics and their diffraction patterns,” Prog. Opt. 18, 258–346 (1980).

M. V. Berry, “Waves and Thom’s theorem,” Adv. Phys. 25, 1–26 (1976).
[CrossRef]

Dangelmayr, G.

G. Dangelmayr, F. J. Wright, “On the validity of the paraxial eikonal in catastrophe optics,” J. Phys. A 17, 99–108 (1984).
[CrossRef]

Dean, C. E.

P. L. Marston, C. E. Dean, H. J. Simpson, “Light scattering from spheroidal drops: exploring optical catastrophes and generalized rainbows,” AIP Conf. Proc. 197, 275–285 (1989). [Because of printing errors, replace (γ/α) in Eq. (4) by (γ/α)1/2 and κ1 in Eq. (6) by κ2.]
[CrossRef]

C. E. Dean, “Analysis of scattered light: II. The opening rate of the transverse cusp from oblate drops,” Ph.D. dissertation (Washington State University, Pullman, Wash., 1989).

Gullstrand, A.

A. Gullstrand, “Die reele optische Abbildung,” Sven. Vetensk. Handl. 41, 1–119 (1906).

Hannay, J. H.

J. F. Nye, J. H. Hannay, “The orientations and distortions of caustics in geometrical optics,” Opt. Acta 31, 115–130 (1984).
[CrossRef]

Kneisly, J. A.

Langley, D. S.

Marston, P. L.

H. J. Simpson, P. L. Marston, “Scattering of white light from levitated oblate water drops near rainbows and other diffraction catastrophes,” Appl. Opt. 30, 3466–3472 (1991).
[CrossRef]

P. L. Marston, C. E. Dean, H. J. Simpson, “Light scattering from spheroidal drops: exploring optical catastrophes and generalized rainbows,” AIP Conf. Proc. 197, 275–285 (1989). [Because of printing errors, replace (γ/α) in Eq. (4) by (γ/α)1/2 and κ1 in Eq. (6) by κ2.]
[CrossRef]

P. L. Marston, “Transverse cusp diffraction catastrophes: some pertinent wavefronts and a Pearcey approximation to the wavefield,” J. Acoust. Soc. Am. 81, 226–232 (1987); J. Acoust. Soc. Am. (E) 83, 1976 (1988).
[CrossRef]

P. L. Marston, “Hyperbolic-umbilic diffraction catastrophes and the tracing of local principal curvatures of wave fronts,” J. Acoust. Soc. Am. Suppl. 80, S73 (1986); “Hyperbolic-umbilic focal sections: the wavefield and the merging of rays at caustic lines,” J. Acoust. Soc. Am. Suppl. 81, S14 (1987).
[CrossRef]

P. L. Marston, “Cusp diffraction catastrophes from spheroids: generalized rainbows and inverse scattering,” Opt. Lett. 10, 588–590 (1985).
[CrossRef] [PubMed]

D. S. Langley, P. L. Marston, “Critical-angle scattering of laser light from bubbles in water: measurements, models, and application to sizing of bubbles,” Appl. Opt. 23, 1044–1054 (1984).
[CrossRef] [PubMed]

P. L. Marston, E. H. Trinh, “Hyperbolic umbilic diffraction catastrophe and rainbow scattering from drops,” Nature (London) 312, 529–531 (1984).
[CrossRef]

P. L. Marston, “Wavefront geometries giving transverse cusp and hyperbolic umbilic foci in acoustic shocks,” in Shock Waves in Condensed Matter 1987, S. C. Schmidt, N. C. Holmes, eds. (Elsevier, Amsterdam, 1988), pp. 203–206.

Mugnai, A.

Nye, J. F.

J. F. Nye, J. H. Hannay, “The orientations and distortions of caustics in geometrical optics,” Opt. Acta 31, 115–130 (1984).
[CrossRef]

J. F. Nye, “Rainbow scattering from spheroidal drops—an explanation of the hyperbolic umbilic foci,” Nature (London) 312, 531–532 (1984).
[CrossRef]

J. F. Nye, “Optical caustics in the near field from liquid drops,” Proc. R. Soc. London Ser. A 361, 21–41 (1978).
[CrossRef]

Pearcey, T.

T. Pearcey, “The structure of an electromagnetic field in the neighborhood of a cusp of a caustic,” Philos. Mag. 37, 311–317 (1946).

Simpson, H. J.

H. J. Simpson, P. L. Marston, “Scattering of white light from levitated oblate water drops near rainbows and other diffraction catastrophes,” Appl. Opt. 30, 3466–3472 (1991).
[CrossRef]

P. L. Marston, C. E. Dean, H. J. Simpson, “Light scattering from spheroidal drops: exploring optical catastrophes and generalized rainbows,” AIP Conf. Proc. 197, 275–285 (1989). [Because of printing errors, replace (γ/α) in Eq. (4) by (γ/α)1/2 and κ1 in Eq. (6) by κ2.]
[CrossRef]

H. J. Simpson, “The lips event for light backscattered from levitated water drops,” M.S. degree project report (Washington State University, Pullman, Wash., 1988).

Stavroudis, O. N.

O. N. Stavroudis, The Optics of Rays, Wavefronts, and Caustics (Academic, New York, 1972), pp. 161–179; R. S. Chang, O. N. Stavroudis, “Generalized ray tracing, caustic surfaces, generalized bending, and the construction of a novel merit function for optical design,” J. Opt. Soc. Am. 70, 976–985 (1980); D. G. Burkhard, D. L. Shealy, “Flux density for ray propagation in geometrical optics,” J. pt. Soc. Am. 63, 299–304 (1973); M. Herzberger, Modern Geometrical Optics (Wiley-Interscience, New York, 1958).
[CrossRef]

Trinh, E. H.

P. L. Marston, E. H. Trinh, “Hyperbolic umbilic diffraction catastrophe and rainbow scattering from drops,” Nature (London) 312, 529–531 (1984).
[CrossRef]

Upstill, C.

M. V. Berry, C. Upstill, “Catastrophe optics: morphologies of caustics and their diffraction patterns,” Prog. Opt. 18, 258–346 (1980).

Wiscombe, W. J.

Wright, F. J.

G. Dangelmayr, F. J. Wright, “On the validity of the paraxial eikonal in catastrophe optics,” J. Phys. A 17, 99–108 (1984).
[CrossRef]

Adv. Phys. (1)

M. V. Berry, “Waves and Thom’s theorem,” Adv. Phys. 25, 1–26 (1976).
[CrossRef]

AIP Conf. Proc. (1)

P. L. Marston, C. E. Dean, H. J. Simpson, “Light scattering from spheroidal drops: exploring optical catastrophes and generalized rainbows,” AIP Conf. Proc. 197, 275–285 (1989). [Because of printing errors, replace (γ/α) in Eq. (4) by (γ/α)1/2 and κ1 in Eq. (6) by κ2.]
[CrossRef]

Appl. Opt. (3)

J. Acoust. Soc. Am. (1)

P. L. Marston, “Transverse cusp diffraction catastrophes: some pertinent wavefronts and a Pearcey approximation to the wavefield,” J. Acoust. Soc. Am. 81, 226–232 (1987); J. Acoust. Soc. Am. (E) 83, 1976 (1988).
[CrossRef]

J. Acoust. Soc. Am. Suppl. (1)

P. L. Marston, “Hyperbolic-umbilic diffraction catastrophes and the tracing of local principal curvatures of wave fronts,” J. Acoust. Soc. Am. Suppl. 80, S73 (1986); “Hyperbolic-umbilic focal sections: the wavefield and the merging of rays at caustic lines,” J. Acoust. Soc. Am. Suppl. 81, S14 (1987).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Phys. A (1)

G. Dangelmayr, F. J. Wright, “On the validity of the paraxial eikonal in catastrophe optics,” J. Phys. A 17, 99–108 (1984).
[CrossRef]

Nature (London) (2)

J. F. Nye, “Rainbow scattering from spheroidal drops—an explanation of the hyperbolic umbilic foci,” Nature (London) 312, 531–532 (1984).
[CrossRef]

P. L. Marston, E. H. Trinh, “Hyperbolic umbilic diffraction catastrophe and rainbow scattering from drops,” Nature (London) 312, 529–531 (1984).
[CrossRef]

Opt. Acta (1)

J. F. Nye, J. H. Hannay, “The orientations and distortions of caustics in geometrical optics,” Opt. Acta 31, 115–130 (1984).
[CrossRef]

Opt. Lett. (1)

Philos. Mag. (1)

T. Pearcey, “The structure of an electromagnetic field in the neighborhood of a cusp of a caustic,” Philos. Mag. 37, 311–317 (1946).

Proc. R. Soc. London Ser. A (1)

J. F. Nye, “Optical caustics in the near field from liquid drops,” Proc. R. Soc. London Ser. A 361, 21–41 (1978).
[CrossRef]

Prog. Opt. (1)

M. V. Berry, C. Upstill, “Catastrophe optics: morphologies of caustics and their diffraction patterns,” Prog. Opt. 18, 258–346 (1980).

Sven. Vetensk. Handl. (1)

A. Gullstrand, “Die reele optische Abbildung,” Sven. Vetensk. Handl. 41, 1–119 (1906).

Other (4)

H. J. Simpson, “The lips event for light backscattered from levitated water drops,” M.S. degree project report (Washington State University, Pullman, Wash., 1988).

P. L. Marston, “Wavefront geometries giving transverse cusp and hyperbolic umbilic foci in acoustic shocks,” in Shock Waves in Condensed Matter 1987, S. C. Schmidt, N. C. Holmes, eds. (Elsevier, Amsterdam, 1988), pp. 203–206.

C. E. Dean, “Analysis of scattered light: II. The opening rate of the transverse cusp from oblate drops,” Ph.D. dissertation (Washington State University, Pullman, Wash., 1989).

O. N. Stavroudis, The Optics of Rays, Wavefronts, and Caustics (Academic, New York, 1972), pp. 161–179; R. S. Chang, O. N. Stavroudis, “Generalized ray tracing, caustic surfaces, generalized bending, and the construction of a novel merit function for optical design,” J. Opt. Soc. Am. 70, 976–985 (1980); D. G. Burkhard, D. L. Shealy, “Flux density for ray propagation in geometrical optics,” J. pt. Soc. Am. 63, 299–304 (1973); M. Herzberger, Modern Geometrical Optics (Wiley-Interscience, New York, 1958).
[CrossRef]

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Figures (7)

Fig. 1
Fig. 1

(a) Profile of an oblate drop and the location of rays that leave the drop when the drop is viewed from the four-ray region. (b) Partitioning of the scattering pattern by cusp and ordinary rainbow caustics.

Fig. 2
Fig. 2

Photograph of the far-field pattern of red laser light scattered from an acoustically levitated drop of water. The angular width displayed is 10°. The photograph is representative of ones discussed in greater detail in Refs. 13. The corresponding Airy and cusp caustics are as shown in Fig. 1(b). The horizontal scattering angle designated by θ in Fig. 3 increases from θ ≈ 137° on the left.

Fig. 3
Fig. 3

Ray diagram of the equatorial section of the drop. If ray 1 is a cusp ray (true only for a particular value of q = D/H), the x, y exit plane is perpendicular to ray 1 as it leaves the drop and is tangent to the equator (+y is out of the page). The initially plane wave front is refracted twice, once after entering and again after leaving the drop and is reflected off the back of the drop. Note that the exit plane shifts with the cusp ray, which varies with q.

Fig. 4
Fig. 4

Wave propagates from the exit plane close to the drop to the distance u,v observation plane. In the problem considered, the wave front near the exit plane propagates to give a cusp caustic in the distant observation plane. The exit plane is selected as discussed in the Fig. 3 caption.

Fig. 5
Fig. 5

Plot of the predicted cusp opening rate d vs D/H along with experimental data. The inset shows the region of experimental data on an enlarged scale. The plain error bars are for directly measured D/H ratios. The open diamonds are for indirectly measured values of D/H (see text). The data support the analysis.

Fig. 6
Fig. 6

Log–log plots of the far-field opening rate vs q = D/H near the transition and lips events. The least-squares line-fit slope for the transition event ≈1. For the lips event, it is ≈1/2. The solid-line segments are the fitted least-squares models. The fitted lines are given by Eqs. (16) with the parameters given in Table II, where σ = T and L. The data points indicated by open squares and the left-hand vertical axis belong to the transition event data. The open diamonds and right-hand vertical axis belong to the lips event.

Fig. 7
Fig. 7

Log–log plot of the far-field opening rate vs q = D/H just above and below the D4+ hyperbolic–umbilic focal section event. The least-squares line-fit slope in both cases ≈ − 2. The solid-line segments are the fitted least-squares models. The fitted lines are given by Eq. (16) with qσ = q4 and the parameters given in Table II, where σ = 4+ or 4. The open squares correspond to the 4+ values, and the open diamonds to the 4 values. The fitted lines on this scale appear to be identical.

Tables (2)

Tables Icon

Table I Opening Rates d. vs Aspect Ratloa

Tables Icon

Table II Log–log Least-Squares Fit to the Opening Rate Expression, Eq. (16), for a Drop of Water with Refractive Index n = 1.332

Equations (79)

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

q T = n ( 2 n 2 - 2 ) - 1 / 2 1.070 ,
q L = [ n / ( 2 n - 2 ) ] 1 / 2 1.416 ,
q 4 = [ 3 n 2 / [ 4 ( n 2 - 1 ) ] } 1 / 2 1.311 ,
θ = 180 ° + 2 i - 4 sin - 1 [ ( sin i ) / n ] .
q = n [ 2 w ( w - w ) ] - 1 / 2 ,
w = ( n 2 - sin 2 i c ) 1 / 2 ,
w = ( 1 - sin 2 i c ) 1 / 2 ,
W ( x , y ) = a 1 x 2 + a 2 y 2 x + a 3 y 2 .
Ψ ( x , y , u , v , z ) = { [ z - W ( x , y ) ] 2 + ( u - x ) 2 + ( v - y ) 2 } 1 / 2 .
Ψ z - W ( x , y ) + z ( U 2 + V 2 ) / 2 - ( x U + y V ) ,
Ψ x = 0 ,
Ψ y = 0 ,
H = 0 ,
H = 2 Ψ x 2 2 Ψ y 2 - ( 2 Ψ x y ) 2 ,
U 3 = - d V 2 ,
d = 4 a 2 / 27 a 1 2 ,
U = U - U c ,
U c = 2 a 1 a 3 / a 2 .
L = L 1 - S L ,
N = N 1 - S N ;
L = μ L + γ L s ,
N = cos 2 i cos 2 r ( μ N + γ cos 2 i N s ) ;
L = L - 2 cos i L s ,
N = N - 2 cos i N s ,
k 1 , 2 ½ ( W x x + W y y ) ± ½ [ ( W x x - W y y ) 2 + 4 ( W x y ) 2 ] 1 / 2 .
k 1 = 2 a 1 ,
k 2 = 2 a 2 x c + 2 a 3 ,
k 1 = 2 a 1 ,
k 2 = 0.
k 2 = 2 a 2 ( x c + Δ x ) + 2 a 3 ,
k 2 = 2 a 2 ( x - x c ) ,             x - x c D 2 .
Δ x R H ( 4 ) Δ θ R H ( 4 ) ( θ - θ c ) R H ( 4 ) ( d θ / d i ) i c ( i - i c ) ,
k 2 i | i = i c = 2 a 2 d θ d i | i c R H ( 4 ) ,
k 2 k 2 i | i = i c ( i - i c ) = 2 a 2 d θ d i | i c R H ( 4 ) ( i - i c ) ,
a 2 = d k 2 d i | i = i c a 1 d θ d i | i = i c .
d = 8 27 d L ( 4 ) / d i ( d θ / d i ) N ( 4 ) ,
d = B σ q - q σ A σ ,             σ = T , L , 4 - , 4 + ,
N = ( cos 2 i / cos 2 r ) [ μ N + ( γ / cos 2 i ) N s ] ,
N = 2 cos r - 2 n cos i D cos 2 r .
R H = D n cos 2 r 2 n cos r - 2 cos i .
R H - S = R H - D cos r ,
N t = 2 n cos r - 2 cos i 2 D cos r cos i - D n cos 2 r ,
N = N t ( 2 / cos r ) N s ,
N = 6 cos i - 2 n cos r 2 D cos r cos i - D n cos 2 r = [ R H - ] - 1 ,
R H - S = R H - D cos r ,
N t = D cos r n cos r - 4 cos i 6 cos i - 2 n cos r .
N = ( cos 2 r / cos 2 i ) [ n N t + ( γ / cos 2 r ) N s ] ,
N = 8 cos i - 4 n cos r D cos i ( n cos r - 4 cos i ) = [ R H - ] - 1 ,
X = D 2 ( 1 - cos i ) .
( 2 a 1 ) - 1 = R H ( 4 ) = R H - X ,
R H ( 4 ) = D 2 [ 1 + cos r cos i sin i 4 sin r cos r - 2 cos r sin i ] .
sin 2 i = ( 4 - n 2 ) / 3 ,
( cos r - 1 n cos i ) D cos r = H 2 2 D ,
sin 2 i c = n 2 [ 1 - q - 4 ( 4 F ) - 1 ] ,
F = n - 2 - 1 + q - 2 ,
L = μ L + γ L s ,
L = cos r - 1 n cos i H 2 / ( 2 D ) = 1 / R V .
L = L t - 2 cos r L s ,
1 L t = R V - S = H 2 / ( 2 D ) cos r - 1 n cos i - D cos r ,
L = H 2 + 4 D cos r ( R V - D cos r ) H 2 ( R V - D cos r ) .
L = μ L t + γ L s ,
1 / L t = R V - S = R V - D cos r ,
L = n [ H 2 + 4 D cos r ( R V - D cos r ) ] ( R V - D cos r ) ( H 2 - 4 D 2 cos 2 r ) - H 2 D cos r - 2 D H 2 ( cos i - n cos r ) .
d L ( 4 ) d i = d d i [ L 1 - X L ] ,
d L ( 4 ) d i = d L d i + ( L ) 2 d Y d i ( 1 - X L ) 2 ,
Y = D 2 ( 1 - cos i c ) - δ D 2 sin i c ,
δ = ɛ + 4 ɛ cos i c n cos r c ,
d L ( 4 ) d i = d L d i .
N ( 4 ) = N 1 - X N ,
d = 8 27 ( d L / d i ) ( 1 - X N ) ( d θ / d i ) N .
a = n [ H 2 + 4 D cos r ( R V - D cos r ) ] ,
b = ( R V - D cos r ) ( H 2 - 4 D 2 cos 2 r ) - H 2 D cos r ,
c = - 2 D H 2 ( cos i - n cos r ) ,
d d R V d i = H 2 2 D ( cos r - 1 n cos i ) - 2 ( - sin r d r d i + 1 n sin i ) ,
d L d i = a b - a b b 2 + c ,
a = 4 n D ( 2 D cos r sin r - sin r ( d r / d i ) R V + d cos r ) ,
b = [ d + D sin r ( d r / d i ) ] ( H 2 - 4 D 2 cos 2 r ) + ( R V - D cos r ) [ - 8 D 2 cos r sin r ( d r / d i ) ] + H 2 D sin r ( d r / d i ) ,
c = 2 D H 2 ( sin i + n sin r d r d i ) .
d θ d i | i c = 2 - 4 cos i c n cos r c .

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