Abstract

We present an iterative tomographic algorithm to reconstruct refractive-index profiles for meridional planes of the lens of the spherical fish eye from measurements of deflection angles of refracted rays. Numerical simulations show that the algorithm allows accuracy up to the fourth decimal place, provided that the refractive index can be regarded as an analytical function of the radial coordinate and the experimental errors are neglected. An experimental demonstration is given by applying the algorithm to retrieve the refractive-index profile of a spherical fish lens. The method is conceptually simple and does not require matching of the index of the surrounding medium to that of the surface of the lens, and the related iterative algorithm rapidly converges.

© 2005 Optical Society of America

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References

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  1. D. Marcuse, “Refractive index determination by the focusing method,” Appl. Opt. 18, 9–13 (1979).
    [CrossRef] [PubMed]
  2. V. I. Vlad, N. Ionescu-Pallas, “New treatment of the focusing method and tomography of the refractive index distribution of inhomogeneous optical components,” Opt. Eng. 35, 1305–1310 (1996).
    [CrossRef]
  3. R. D. Fernald, “The optical system of the fish,” in The Visual System of Fish, R. H. Douglas, M. B. A. Djamgoz, eds. (Chapman & Hall, Cambridge, UK, 1990), pp. 50–52.
  4. W. S. Jagger, “The optics of the spherical fish lens,” Vision Res. 32, 1271–1284 (1992).
    [CrossRef] [PubMed]
  5. J. G. Sivak, “Optical variability of the fish lens,” in The Visual System of Fish, R. H. Douglas, M. B. A. Djamgoz, eds. (Chapman & Hall, Cambridge, UK, 1990), pp. 63–74.
  6. P. L. Chu, “Nondestructive measurements of index profile of an optical fibre preform,” Electron. Lett. 13, 736–738 (1977).
    [CrossRef]
  7. K. F. Barrell, C. Pask, “Nondestructive index profile measurements of noncircular optical fibre preforms,” Opt. Commun. 27, 230–234 (1978).
    [CrossRef]
  8. M. C. W. Campbell, “Measurement of refractive index in an intact crystalline lens,” Vision Res. 24, 409–415 (1984).
    [CrossRef] [PubMed]
  9. D. Y. C. Chan, J. P. Ennis, B. K. Pierscionek, G. Smith, “Determination and modeling of the 3-D gradient refractive indices in crystalline lenses,” Appl. Opt. 27, 926–931 (1988).
    [CrossRef] [PubMed]
  10. B. K. Pierscionek, D. Y. C. Chan, “Refractive index gradient of human lenses,” Optom. Vision Sci. 66, 822–829 (1989).
    [CrossRef]
  11. R. H. H. Kröger, M. C. W. Campbell, R. Munger, R. D. Fernald, “Refractive index distribution and spherical aberration in the crystalline lens of the African cichlid fish Haplochromis burtoni,” Vision Res. 34, 1815–1822 (1994).
    [CrossRef] [PubMed]
  12. G. Smith, B. K. Pierscionek, D. A. Atchison, “The optical modelling of the human lens,” Ophthalmic Physiol. Opt. 11, 359–369 (1991).
    [CrossRef] [PubMed]
  13. L. F. Garner, G. Smith, S. Yao, R. C. Augusteyn, “Gradient refractive index of the crystalline lens of the Black Oreo Dory (Allocyttus Niger): comparison of magnetic resonance imaging (MRI) and laser ray-trace methods,” Vision Res. 41, 973–979 (2001).
    [CrossRef] [PubMed]
  14. E. Acosta, R. Flores, D. Vazquez, S. Rios, L. F. Garner, G. Smith, “Tomographic method for measurement of the refractive index profile of optical fibre preforms and rod GRIN lenses,” Jpn. J. Appl. Phys. 41, 4821–4824 (2002).
    [CrossRef]
  15. G. T. Herman, Image Reconstruction by Projections (The Fundamentals of Computerized Tomography) (Academic, London, 1980), Chap. 6, “Basic concepts of reconstruction algorithms,” pp. 90–107.
  16. E. W. Marchand, Gradient Index Optics (Academic, New York, 1978), Chap. 2, “Spherical gradients,” pp. 7–9.
  17. D. T. Moore, “Ray tracing in gradient-index media,” J. Opt. Soc. Am. 65, 451–455 (1975).
    [CrossRef]
  18. Ref. 16, Chap. 6, “Ray tracing in a radial gradient,” p. 67.
  19. A. M. Cormack, “Representation of a function by its line integrals, with some radiological applications. II,” J. Appl. Phys. 35, 2908–2912 (1964).
    [CrossRef]
  20. A. Sharma, D. Vizia Kumar, A. K. Ghatak, “Tracing rays through graded-index media: a new method,” Appl. Opt. 21, 984–987 (1982).
    [CrossRef] [PubMed]
  21. D. Axelrod, D. Lerner, P. J. Sands, “Refractive index within the lens of a goldfish eye determined from the paths of thin laser beams,” Vision Res. 28, 57–65 (1988).
    [CrossRef] [PubMed]
  22. M. C. W. Campbell, P. J. Sands, “Optical quality during crystalline lens growth,” Nature 312, 291–292 (1984).
    [CrossRef] [PubMed]

2002 (1)

E. Acosta, R. Flores, D. Vazquez, S. Rios, L. F. Garner, G. Smith, “Tomographic method for measurement of the refractive index profile of optical fibre preforms and rod GRIN lenses,” Jpn. J. Appl. Phys. 41, 4821–4824 (2002).
[CrossRef]

2001 (1)

L. F. Garner, G. Smith, S. Yao, R. C. Augusteyn, “Gradient refractive index of the crystalline lens of the Black Oreo Dory (Allocyttus Niger): comparison of magnetic resonance imaging (MRI) and laser ray-trace methods,” Vision Res. 41, 973–979 (2001).
[CrossRef] [PubMed]

1996 (1)

V. I. Vlad, N. Ionescu-Pallas, “New treatment of the focusing method and tomography of the refractive index distribution of inhomogeneous optical components,” Opt. Eng. 35, 1305–1310 (1996).
[CrossRef]

1994 (1)

R. H. H. Kröger, M. C. W. Campbell, R. Munger, R. D. Fernald, “Refractive index distribution and spherical aberration in the crystalline lens of the African cichlid fish Haplochromis burtoni,” Vision Res. 34, 1815–1822 (1994).
[CrossRef] [PubMed]

1992 (1)

W. S. Jagger, “The optics of the spherical fish lens,” Vision Res. 32, 1271–1284 (1992).
[CrossRef] [PubMed]

1991 (1)

G. Smith, B. K. Pierscionek, D. A. Atchison, “The optical modelling of the human lens,” Ophthalmic Physiol. Opt. 11, 359–369 (1991).
[CrossRef] [PubMed]

1989 (1)

B. K. Pierscionek, D. Y. C. Chan, “Refractive index gradient of human lenses,” Optom. Vision Sci. 66, 822–829 (1989).
[CrossRef]

1988 (2)

D. Axelrod, D. Lerner, P. J. Sands, “Refractive index within the lens of a goldfish eye determined from the paths of thin laser beams,” Vision Res. 28, 57–65 (1988).
[CrossRef] [PubMed]

D. Y. C. Chan, J. P. Ennis, B. K. Pierscionek, G. Smith, “Determination and modeling of the 3-D gradient refractive indices in crystalline lenses,” Appl. Opt. 27, 926–931 (1988).
[CrossRef] [PubMed]

1984 (2)

M. C. W. Campbell, P. J. Sands, “Optical quality during crystalline lens growth,” Nature 312, 291–292 (1984).
[CrossRef] [PubMed]

M. C. W. Campbell, “Measurement of refractive index in an intact crystalline lens,” Vision Res. 24, 409–415 (1984).
[CrossRef] [PubMed]

1982 (1)

1979 (1)

1978 (1)

K. F. Barrell, C. Pask, “Nondestructive index profile measurements of noncircular optical fibre preforms,” Opt. Commun. 27, 230–234 (1978).
[CrossRef]

1977 (1)

P. L. Chu, “Nondestructive measurements of index profile of an optical fibre preform,” Electron. Lett. 13, 736–738 (1977).
[CrossRef]

1975 (1)

1964 (1)

A. M. Cormack, “Representation of a function by its line integrals, with some radiological applications. II,” J. Appl. Phys. 35, 2908–2912 (1964).
[CrossRef]

Acosta, E.

E. Acosta, R. Flores, D. Vazquez, S. Rios, L. F. Garner, G. Smith, “Tomographic method for measurement of the refractive index profile of optical fibre preforms and rod GRIN lenses,” Jpn. J. Appl. Phys. 41, 4821–4824 (2002).
[CrossRef]

Atchison, D. A.

G. Smith, B. K. Pierscionek, D. A. Atchison, “The optical modelling of the human lens,” Ophthalmic Physiol. Opt. 11, 359–369 (1991).
[CrossRef] [PubMed]

Augusteyn, R. C.

L. F. Garner, G. Smith, S. Yao, R. C. Augusteyn, “Gradient refractive index of the crystalline lens of the Black Oreo Dory (Allocyttus Niger): comparison of magnetic resonance imaging (MRI) and laser ray-trace methods,” Vision Res. 41, 973–979 (2001).
[CrossRef] [PubMed]

Axelrod, D.

D. Axelrod, D. Lerner, P. J. Sands, “Refractive index within the lens of a goldfish eye determined from the paths of thin laser beams,” Vision Res. 28, 57–65 (1988).
[CrossRef] [PubMed]

Barrell, K. F.

K. F. Barrell, C. Pask, “Nondestructive index profile measurements of noncircular optical fibre preforms,” Opt. Commun. 27, 230–234 (1978).
[CrossRef]

Campbell, M. C. W.

R. H. H. Kröger, M. C. W. Campbell, R. Munger, R. D. Fernald, “Refractive index distribution and spherical aberration in the crystalline lens of the African cichlid fish Haplochromis burtoni,” Vision Res. 34, 1815–1822 (1994).
[CrossRef] [PubMed]

M. C. W. Campbell, “Measurement of refractive index in an intact crystalline lens,” Vision Res. 24, 409–415 (1984).
[CrossRef] [PubMed]

M. C. W. Campbell, P. J. Sands, “Optical quality during crystalline lens growth,” Nature 312, 291–292 (1984).
[CrossRef] [PubMed]

Chan, D. Y. C.

Chu, P. L.

P. L. Chu, “Nondestructive measurements of index profile of an optical fibre preform,” Electron. Lett. 13, 736–738 (1977).
[CrossRef]

Cormack, A. M.

A. M. Cormack, “Representation of a function by its line integrals, with some radiological applications. II,” J. Appl. Phys. 35, 2908–2912 (1964).
[CrossRef]

Ennis, J. P.

Fernald, R. D.

R. H. H. Kröger, M. C. W. Campbell, R. Munger, R. D. Fernald, “Refractive index distribution and spherical aberration in the crystalline lens of the African cichlid fish Haplochromis burtoni,” Vision Res. 34, 1815–1822 (1994).
[CrossRef] [PubMed]

R. D. Fernald, “The optical system of the fish,” in The Visual System of Fish, R. H. Douglas, M. B. A. Djamgoz, eds. (Chapman & Hall, Cambridge, UK, 1990), pp. 50–52.

Flores, R.

E. Acosta, R. Flores, D. Vazquez, S. Rios, L. F. Garner, G. Smith, “Tomographic method for measurement of the refractive index profile of optical fibre preforms and rod GRIN lenses,” Jpn. J. Appl. Phys. 41, 4821–4824 (2002).
[CrossRef]

Garner, L. F.

E. Acosta, R. Flores, D. Vazquez, S. Rios, L. F. Garner, G. Smith, “Tomographic method for measurement of the refractive index profile of optical fibre preforms and rod GRIN lenses,” Jpn. J. Appl. Phys. 41, 4821–4824 (2002).
[CrossRef]

L. F. Garner, G. Smith, S. Yao, R. C. Augusteyn, “Gradient refractive index of the crystalline lens of the Black Oreo Dory (Allocyttus Niger): comparison of magnetic resonance imaging (MRI) and laser ray-trace methods,” Vision Res. 41, 973–979 (2001).
[CrossRef] [PubMed]

Ghatak, A. K.

Herman, G. T.

G. T. Herman, Image Reconstruction by Projections (The Fundamentals of Computerized Tomography) (Academic, London, 1980), Chap. 6, “Basic concepts of reconstruction algorithms,” pp. 90–107.

Ionescu-Pallas, N.

V. I. Vlad, N. Ionescu-Pallas, “New treatment of the focusing method and tomography of the refractive index distribution of inhomogeneous optical components,” Opt. Eng. 35, 1305–1310 (1996).
[CrossRef]

Jagger, W. S.

W. S. Jagger, “The optics of the spherical fish lens,” Vision Res. 32, 1271–1284 (1992).
[CrossRef] [PubMed]

Kröger, R. H. H.

R. H. H. Kröger, M. C. W. Campbell, R. Munger, R. D. Fernald, “Refractive index distribution and spherical aberration in the crystalline lens of the African cichlid fish Haplochromis burtoni,” Vision Res. 34, 1815–1822 (1994).
[CrossRef] [PubMed]

Lerner, D.

D. Axelrod, D. Lerner, P. J. Sands, “Refractive index within the lens of a goldfish eye determined from the paths of thin laser beams,” Vision Res. 28, 57–65 (1988).
[CrossRef] [PubMed]

Marchand, E. W.

E. W. Marchand, Gradient Index Optics (Academic, New York, 1978), Chap. 2, “Spherical gradients,” pp. 7–9.

Marcuse, D.

Moore, D. T.

Munger, R.

R. H. H. Kröger, M. C. W. Campbell, R. Munger, R. D. Fernald, “Refractive index distribution and spherical aberration in the crystalline lens of the African cichlid fish Haplochromis burtoni,” Vision Res. 34, 1815–1822 (1994).
[CrossRef] [PubMed]

Pask, C.

K. F. Barrell, C. Pask, “Nondestructive index profile measurements of noncircular optical fibre preforms,” Opt. Commun. 27, 230–234 (1978).
[CrossRef]

Pierscionek, B. K.

G. Smith, B. K. Pierscionek, D. A. Atchison, “The optical modelling of the human lens,” Ophthalmic Physiol. Opt. 11, 359–369 (1991).
[CrossRef] [PubMed]

B. K. Pierscionek, D. Y. C. Chan, “Refractive index gradient of human lenses,” Optom. Vision Sci. 66, 822–829 (1989).
[CrossRef]

D. Y. C. Chan, J. P. Ennis, B. K. Pierscionek, G. Smith, “Determination and modeling of the 3-D gradient refractive indices in crystalline lenses,” Appl. Opt. 27, 926–931 (1988).
[CrossRef] [PubMed]

Rios, S.

E. Acosta, R. Flores, D. Vazquez, S. Rios, L. F. Garner, G. Smith, “Tomographic method for measurement of the refractive index profile of optical fibre preforms and rod GRIN lenses,” Jpn. J. Appl. Phys. 41, 4821–4824 (2002).
[CrossRef]

Sands, P. J.

D. Axelrod, D. Lerner, P. J. Sands, “Refractive index within the lens of a goldfish eye determined from the paths of thin laser beams,” Vision Res. 28, 57–65 (1988).
[CrossRef] [PubMed]

M. C. W. Campbell, P. J. Sands, “Optical quality during crystalline lens growth,” Nature 312, 291–292 (1984).
[CrossRef] [PubMed]

Sharma, A.

Sivak, J. G.

J. G. Sivak, “Optical variability of the fish lens,” in The Visual System of Fish, R. H. Douglas, M. B. A. Djamgoz, eds. (Chapman & Hall, Cambridge, UK, 1990), pp. 63–74.

Smith, G.

E. Acosta, R. Flores, D. Vazquez, S. Rios, L. F. Garner, G. Smith, “Tomographic method for measurement of the refractive index profile of optical fibre preforms and rod GRIN lenses,” Jpn. J. Appl. Phys. 41, 4821–4824 (2002).
[CrossRef]

L. F. Garner, G. Smith, S. Yao, R. C. Augusteyn, “Gradient refractive index of the crystalline lens of the Black Oreo Dory (Allocyttus Niger): comparison of magnetic resonance imaging (MRI) and laser ray-trace methods,” Vision Res. 41, 973–979 (2001).
[CrossRef] [PubMed]

G. Smith, B. K. Pierscionek, D. A. Atchison, “The optical modelling of the human lens,” Ophthalmic Physiol. Opt. 11, 359–369 (1991).
[CrossRef] [PubMed]

D. Y. C. Chan, J. P. Ennis, B. K. Pierscionek, G. Smith, “Determination and modeling of the 3-D gradient refractive indices in crystalline lenses,” Appl. Opt. 27, 926–931 (1988).
[CrossRef] [PubMed]

Vazquez, D.

E. Acosta, R. Flores, D. Vazquez, S. Rios, L. F. Garner, G. Smith, “Tomographic method for measurement of the refractive index profile of optical fibre preforms and rod GRIN lenses,” Jpn. J. Appl. Phys. 41, 4821–4824 (2002).
[CrossRef]

Vizia Kumar, D.

Vlad, V. I.

V. I. Vlad, N. Ionescu-Pallas, “New treatment of the focusing method and tomography of the refractive index distribution of inhomogeneous optical components,” Opt. Eng. 35, 1305–1310 (1996).
[CrossRef]

Yao, S.

L. F. Garner, G. Smith, S. Yao, R. C. Augusteyn, “Gradient refractive index of the crystalline lens of the Black Oreo Dory (Allocyttus Niger): comparison of magnetic resonance imaging (MRI) and laser ray-trace methods,” Vision Res. 41, 973–979 (2001).
[CrossRef] [PubMed]

Appl. Opt. (3)

Electron. Lett. (1)

P. L. Chu, “Nondestructive measurements of index profile of an optical fibre preform,” Electron. Lett. 13, 736–738 (1977).
[CrossRef]

J. Appl. Phys. (1)

A. M. Cormack, “Representation of a function by its line integrals, with some radiological applications. II,” J. Appl. Phys. 35, 2908–2912 (1964).
[CrossRef]

J. Opt. Soc. Am. (1)

Jpn. J. Appl. Phys. (1)

E. Acosta, R. Flores, D. Vazquez, S. Rios, L. F. Garner, G. Smith, “Tomographic method for measurement of the refractive index profile of optical fibre preforms and rod GRIN lenses,” Jpn. J. Appl. Phys. 41, 4821–4824 (2002).
[CrossRef]

Nature (1)

M. C. W. Campbell, P. J. Sands, “Optical quality during crystalline lens growth,” Nature 312, 291–292 (1984).
[CrossRef] [PubMed]

Ophthalmic Physiol. Opt. (1)

G. Smith, B. K. Pierscionek, D. A. Atchison, “The optical modelling of the human lens,” Ophthalmic Physiol. Opt. 11, 359–369 (1991).
[CrossRef] [PubMed]

Opt. Commun. (1)

K. F. Barrell, C. Pask, “Nondestructive index profile measurements of noncircular optical fibre preforms,” Opt. Commun. 27, 230–234 (1978).
[CrossRef]

Opt. Eng. (1)

V. I. Vlad, N. Ionescu-Pallas, “New treatment of the focusing method and tomography of the refractive index distribution of inhomogeneous optical components,” Opt. Eng. 35, 1305–1310 (1996).
[CrossRef]

Optom. Vision Sci. (1)

B. K. Pierscionek, D. Y. C. Chan, “Refractive index gradient of human lenses,” Optom. Vision Sci. 66, 822–829 (1989).
[CrossRef]

Vision Res. (5)

R. H. H. Kröger, M. C. W. Campbell, R. Munger, R. D. Fernald, “Refractive index distribution and spherical aberration in the crystalline lens of the African cichlid fish Haplochromis burtoni,” Vision Res. 34, 1815–1822 (1994).
[CrossRef] [PubMed]

M. C. W. Campbell, “Measurement of refractive index in an intact crystalline lens,” Vision Res. 24, 409–415 (1984).
[CrossRef] [PubMed]

W. S. Jagger, “The optics of the spherical fish lens,” Vision Res. 32, 1271–1284 (1992).
[CrossRef] [PubMed]

L. F. Garner, G. Smith, S. Yao, R. C. Augusteyn, “Gradient refractive index of the crystalline lens of the Black Oreo Dory (Allocyttus Niger): comparison of magnetic resonance imaging (MRI) and laser ray-trace methods,” Vision Res. 41, 973–979 (2001).
[CrossRef] [PubMed]

D. Axelrod, D. Lerner, P. J. Sands, “Refractive index within the lens of a goldfish eye determined from the paths of thin laser beams,” Vision Res. 28, 57–65 (1988).
[CrossRef] [PubMed]

Other (5)

G. T. Herman, Image Reconstruction by Projections (The Fundamentals of Computerized Tomography) (Academic, London, 1980), Chap. 6, “Basic concepts of reconstruction algorithms,” pp. 90–107.

E. W. Marchand, Gradient Index Optics (Academic, New York, 1978), Chap. 2, “Spherical gradients,” pp. 7–9.

Ref. 16, Chap. 6, “Ray tracing in a radial gradient,” p. 67.

J. G. Sivak, “Optical variability of the fish lens,” in The Visual System of Fish, R. H. Douglas, M. B. A. Djamgoz, eds. (Chapman & Hall, Cambridge, UK, 1990), pp. 63–74.

R. D. Fernald, “The optical system of the fish,” in The Visual System of Fish, R. H. Douglas, M. B. A. Djamgoz, eds. (Chapman & Hall, Cambridge, UK, 1990), pp. 50–52.

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Figures (10)

Fig. 1
Fig. 1

Tomographic reconstruction from (a) the measurement of the attenuation At ( x ) for a density object of absorption distribution α ( x ,   z ) through x rays, which follow straight trajectories, (b) the measurement of the optical path S ( x ,   R ) for an inhomogeneous medium of a circular section of radius R through light rays. Their trajectories are, in this case, a function of the inner distribution of refractive index n ( x ,   z ) .

Fig. 2
Fig. 2

A zero-slope incoming ray deflected in the meridional plane of a spherical fish lens.

Fig. 3
Fig. 3

Rotation of the coordinate system, θ = arctan [ ( x C - x B ) / ( z C - z B ) ] .

Fig. 4
Fig. 4

Trajectory of a given ray in the parabolic approximation for the rotated system X Z .

Fig. 5
Fig. 5

Difference between the retrieved and the theoretical gradient indices for three apertures: 0.8 R (dotted curve), 0.9 R (dashed–dotted curve), and 1.0 R (solid curve).

Fig. 6
Fig. 6

rms of the difference between the retrieved and the theoretical gradient indices for 5000 cases, assuming for the sine distribution a Gaussian error zero mean and a standard deviation of 10 - 3 .

Fig. 7
Fig. 7

Assembled image of a bundle of incoming laser beams traversing a fish lens.

Fig. 8
Fig. 8

Comparison of the retrieved (solid curve) and experimental (dotted curve) distributions of the sines of the deflection angles at z = R for the sampled section of the lens, which corresponded to an aperture of 0.9 R . The standard deviation between both distributions is 3 × 10 - 3 .

Fig. 9
Fig. 9

Three refractive-index distributions represented as a solid curve for n 1 ( r ) , a dashed curve for n 2 ( r ) , and a dashed–dotted curve for n 3 ( r ) .

Fig. 10
Fig. 10

Rms of the difference between the theoretical and the retrieved gradient indices for n 1 ( r ) , n 2 ( r ) , and n 3 ( r ) for an aperture of 0.9 R . Four results are given: The two first are those calculated at the straight and parabolic stages of the usual tomographic algorithm and the two following ones are those calculated through parabolic rays for two given initial surface indices, n s = 1.36 and n s = 1.37 .

Tables (1)

Tables Icon

Table 1 Results of the Tomographic Algorithm for Three Apertures

Equations (40)

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At ( P 1 P 2 ) = P 1 P 2 α ( x ,   z ) d s ,
S ( P 1 P 2 ) = P 1 P 2 n ( x ,   z ) d s ,
n ext   d r d s = L ˆ ( x ,   z ) ,
n ext sin α ( x ,   R ) = L ˆ ( x ,   R ) x .
L ˆ ( x ,   R ) = n ext 0 x sin α ( x ,   R ) d x + L ˆ ( 0 ,   R ) .
S ( AD ) = L ˆ ( D ) - L ˆ ( A )
S ( A o D o ) = L ˆ ( D o ) - L ˆ ( A o ) .
L ˆ ( A ) = L ˆ ( A o ) ,
S ( AD ) = S m ( AD ) + K ,
S m ( AD ) = n ext AB ¯ + S ( BC ) + n ext CD ¯ .
x B = - R sin β ,
z B = - R cos β ,
β = α - arctan ( x C / z C ) .
S ( BC ) = B C n ( r ) d s ,
S ( BC ) = B C n ( r ) d s = S m ( AD ) + K - n ext ( AB ¯ + CD ¯ ) ;
S m ( BC ) S m ( AD ) - n ext ( AB ¯ + CD ¯ ) = B C n ( r ) d s - K .
S ( AD ) = - R x sin α ( x ,   R ) d x + K ,
n ( x 2 + z 2 ) = n ( r ) = j = 0 M a j r 2 j ,
S m ( BC ) = B C n ( r ) d s - K .
S m ( BC ) = - R 2 - x 2 R 2 - x 2 n ( x 2 + z 2 ) d z - K ,
S m ( BC ) = - R 2 - x 2 R 2 - x 2 j = 0 M a j ( x 2 + z 2 ) j d z - K ,
S m ( BC ) = j = 0 M a j - R 2 - x 2 R 2 - x 2 ( x 2 + z 2 ) j d z - K ,
- R 2 - x 2 R 2 - x 2 g ( x ,   z ) d z = K , x [ 0 ,   R ]
g ( x ,   z ) = K R 2 - x 2 - z 2 ,
S m ( BC ) = - R 2 - x 2 R 2 - x 2 n ( x ,   z ) - K R 2 - x 2 - z 2 d z .
n ( x ,   z ) - K R 2 - x 2 - z 2 ,
l = 1 N j = 0 M a j 0 f j l - K - S m ( B l C l ) 2 ,
f j l = - R 2 - c l 2 R 2 - c l 2 ( c l 2 + z 2 ) j d z .
tan ϕ = 2 c 2 z c = d x d z ,
ϕ = β + φ = arcsin n ext sin β n s 0 + arctan x c z c .
c 1 = x c - c 2 z c 2 , c 2 = tan ϕ 2 z c .
S m ( BC ) = - R 2 - x 2 R 2 - x 2 [ n ( x ,   z ) - g ( x ,   z ) ] d s ,
l = 1 N j = 0 M a j 1 h j l - K - S m ( B l C l ) 2 ,
h j l = - R 2 - x l 2 R 2 - x l 2 [ ( c 1 l + c 2 l z 2 ) 2 + z 2 ] j [ 1 + ( 2 c 2 l z ) 2 ] 1 / 2 d z
B ( x 0 ) C ( x 0 ) Δ n ( r ) d s n ext 0 x 0 Δ [ sin α ( x ) ] d x .
Δ n 2 R n ext Δ ( sin α ) R .
Δ n n ext 2   Δ ( sin α ) .
n ( r ) = j = 0 3 a j r R 2 j ,
rms = 0 1 [ n r ( r ) - n t ( r ) ] 2 d r 1 / 2 .
sin α ( x ) = i = 1 N c i x ( 2 i - 1 )

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