Abstract

The gradient index (GRIN) model is the most accurate way to represent the eye lens which, because of its growth mode, is a lamellar, shell-like structure. The GRIN is thought to provide optical properties that counteract age-related changes in curvature that would otherwise create an increasingly myopic eye: the so-called lens paradox. This article investigates how fine-tuning the refractive index and the internal curvatures of the lenticular indicial contours may prevent the ageing eye from becoming myopic. A system matrix approach is applied for analysis of a shell model with 200 shells to obtain the paraxial characteristics of the eye model.

© 2017 Optical Society of America

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References

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  4. B. Pierscionek, M. Bahrami, M. Hoshino, K. Uesugi, J. Regini, and N. Yagi, “The eye lens: age-related trends and individual variations in refractive index and shape parameters,” Oncotarget 6(31), 30532–30544 (2015).
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  18. G. Smith and B. K. Pierscionek, “The optical structure of the lens and its contribution to the refractive status of the eye,” Ophthalmic Physiol. Opt. 18(1), 21–29 (1998).
    [Crossref] [PubMed]
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  25. T. Evans and W. F. Harris, “Dependence of the transference of a reduced eye on frequency of light,” S. Afr. Optom. 70, 149–155 (2011).
  26. C. E. Jones, D. A. Atchison, R. Meder, and J. M. Pope, “Refractive index distribution and optical properties of the isolated human lens measured using magnetic resonance imaging (MRI),” Vision Res. 45(18), 2352–2366 (2005).
    [Crossref] [PubMed]
  27. R. Navarro and N. Lopez-Gil, “Impact of the internal curvature gradient on the power and accommodation of the crystalline lens,” Optica 4(3), 334–340 (2017).
    [Crossref]
  28. G. Smith, D. A. Atchison, and B. K. Pierscionek, “Modelling the ageing human eye,” J. Opt. Soc. Am. A 9(12), 2111–2117 (1992).
    [Crossref]
  29. B. K. Pierscionek and J. W. Regini, “The gradient index lens of the eye: an opto-biological synchrony,” Prog. Retin. Eye Res. 31(4), 332–349 (2012).
    [Crossref] [PubMed]
  30. D. Lahm, L. K. Lee, and F. A. Bettelheim, “Age dependence of freezable and nonfreezable water content of normal human lenses,” Invest. Ophthalmol. Vis. Sci. 26(8), 1162–1165 (1985).
    [PubMed]
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    [Crossref] [PubMed]
  32. W. N. Charman, Adnan, and D. A. Atchison, “Gradients of refractive index in the crystalline lens and transient changes in refraction among patients with diabetes,” Biomed. Opt. Express 3(12), 3033–3042 (2012).
    [PubMed]
  33. S. F. Lin, P. K. Lin, F. L. Chang, and R. K. Tsai, “Transient hyperopia after intensive treatment of hyperglycemia in newly diagnosed diabetes,” Ophthalmologica 223(1), 68–71 (2009).
    [Crossref] [PubMed]
  34. B. E. Klein, K. E. Lee, and R. Klein, “Refraction in adults with diabetes,” Arch. Ophthalmol. 129(1), 56–62 (2011).
    [Crossref] [PubMed]

2017 (1)

2016 (1)

2015 (1)

B. Pierscionek, M. Bahrami, M. Hoshino, K. Uesugi, J. Regini, and N. Yagi, “The eye lens: age-related trends and individual variations in refractive index and shape parameters,” Oncotarget 6(31), 30532–30544 (2015).
[Crossref] [PubMed]

2012 (2)

2011 (2)

B. E. Klein, K. E. Lee, and R. Klein, “Refraction in adults with diabetes,” Arch. Ophthalmol. 129(1), 56–62 (2011).
[Crossref] [PubMed]

T. Evans and W. F. Harris, “Dependence of the transference of a reduced eye on frequency of light,” S. Afr. Optom. 70, 149–155 (2011).

2009 (1)

S. F. Lin, P. K. Lin, F. L. Chang, and R. K. Tsai, “Transient hyperopia after intensive treatment of hyperglycemia in newly diagnosed diabetes,” Ophthalmologica 223(1), 68–71 (2009).
[Crossref] [PubMed]

2008 (1)

D. A. Atchison, E. L. Markwell, S. Kasthurirangan, J. M. Pope, G. Smith, and P. G. Swann, “Age-related changes in optical and biometric characteristics of emmetropic eyes,” J. Vis. 8(4), 29 (2008).
[Crossref] [PubMed]

2007 (2)

2006 (1)

M. Dubbelman, V. A. Sicam, and G. L. Van der Heijde, “The shape of the anterior and posterior surface of the aging human cornea,” Vision Res. 46(6-7), 993–1001 (2006).
[Crossref] [PubMed]

2005 (2)

S. Norrby, “The Dubbelman eye model analysed by ray tracing through aspheric surfaces,” Ophthalmic Physiol. Opt. 25(2), 153–161 (2005).
[Crossref] [PubMed]

C. E. Jones, D. A. Atchison, R. Meder, and J. M. Pope, “Refractive index distribution and optical properties of the isolated human lens measured using magnetic resonance imaging (MRI),” Vision Res. 45(18), 2352–2366 (2005).
[Crossref] [PubMed]

2001 (3)

W. F. Harris, “Magnification, blur, and ray state at the retina for the general eye with and without a general optical instrument in front of it: 1. Distant Objects,” Optom. Vis. Sci. 78(12), 888–900 (2001).
[Crossref] [PubMed]

M. Dubbelman, G. L. van der Heijde, and H. A. Weeber, “The thickness of the aging human lens obtained from corrected Scheimpflug images,” Optom. Vis. Sci. 78(6), 411–416 (2001).
[Crossref] [PubMed]

M. Dubbelman and G. L. Van der Heijde, “The shape of the aging human lens: Curvature, equivalent refractive index and the lens paradox,” Vision Res. 41(14), 1867–1877 (2001).
[Crossref] [PubMed]

1998 (1)

G. Smith and B. K. Pierscionek, “The optical structure of the lens and its contribution to the refractive status of the eye,” Ophthalmic Physiol. Opt. 18(1), 21–29 (1998).
[Crossref] [PubMed]

1997 (1)

1992 (1)

1990 (1)

B. Pierscionek, “Presbyopia–effect of refractive index,” Clin. Exp. Optom. 73(1), 23–30 (1990).
[Crossref]

1985 (2)

D. Lahm, L. K. Lee, and F. A. Bettelheim, “Age dependence of freezable and nonfreezable water content of normal human lenses,” Invest. Ophthalmol. Vis. Sci. 26(8), 1162–1165 (1985).
[PubMed]

R. Navarro, J. Santamaría, and J. Bescós, “Accommodation-dependent model of the human eye with aspherics,” J. Opt. Soc. Am. A 2(8), 1273–1281 (1985).
[Crossref] [PubMed]

1984 (1)

O. Pomerantzeff, M. Pankratov, G. J. Wang, and P. Dufault, “Wide-angle optical model of the eye,” Am. J. Optom. Physiol. Opt. 61(3), 166–176 (1984).
[Crossref] [PubMed]

1983 (1)

1982 (1)

P. R. Eva, P. T. Pascoe, and D. G. Vaughan, “Refractive change in hyperglycaemia: hyperopia, not myopia,” Br. J. Ophthalmol. 66(8), 500–505 (1982).
[Crossref] [PubMed]

1974 (2)

N. Brown, “The change in lens curvature with age,” Exp. Eye Res. 19(2), 175–183 (1974).
[Crossref] [PubMed]

N. Drasdo and C. W. Fowler, “Non-linear projection of the retinal image in a wide-angle schematic eye,” Br. J. Ophthalmol. 58(8), 709–714 (1974).
[Crossref] [PubMed]

1971 (1)

Adnan,

Atchison, D. A.

W. N. Charman, Adnan, and D. A. Atchison, “Gradients of refractive index in the crystalline lens and transient changes in refraction among patients with diabetes,” Biomed. Opt. Express 3(12), 3033–3042 (2012).
[PubMed]

D. A. Atchison, E. L. Markwell, S. Kasthurirangan, J. M. Pope, G. Smith, and P. G. Swann, “Age-related changes in optical and biometric characteristics of emmetropic eyes,” J. Vis. 8(4), 29 (2008).
[Crossref] [PubMed]

C. E. Jones, D. A. Atchison, R. Meder, and J. M. Pope, “Refractive index distribution and optical properties of the isolated human lens measured using magnetic resonance imaging (MRI),” Vision Res. 45(18), 2352–2366 (2005).
[Crossref] [PubMed]

G. Smith, D. A. Atchison, and B. K. Pierscionek, “Modelling the ageing human eye,” J. Opt. Soc. Am. A 9(12), 2111–2117 (1992).
[Crossref]

Bahrami, M.

B. Pierscionek, M. Bahrami, M. Hoshino, K. Uesugi, J. Regini, and N. Yagi, “The eye lens: age-related trends and individual variations in refractive index and shape parameters,” Oncotarget 6(31), 30532–30544 (2015).
[Crossref] [PubMed]

Bescós, J.

Bettelheim, F. A.

D. Lahm, L. K. Lee, and F. A. Bettelheim, “Age dependence of freezable and nonfreezable water content of normal human lenses,” Invest. Ophthalmol. Vis. Sci. 26(8), 1162–1165 (1985).
[PubMed]

Brennan, N. A.

Brown, N.

N. Brown, “The change in lens curvature with age,” Exp. Eye Res. 19(2), 175–183 (1974).
[Crossref] [PubMed]

Chang, F. L.

S. F. Lin, P. K. Lin, F. L. Chang, and R. K. Tsai, “Transient hyperopia after intensive treatment of hyperglycemia in newly diagnosed diabetes,” Ophthalmologica 223(1), 68–71 (2009).
[Crossref] [PubMed]

Charman, W. N.

Dainty, C.

Drasdo, N.

N. Drasdo and C. W. Fowler, “Non-linear projection of the retinal image in a wide-angle schematic eye,” Br. J. Ophthalmol. 58(8), 709–714 (1974).
[Crossref] [PubMed]

Dubbelman, M.

M. Dubbelman, V. A. Sicam, and G. L. Van der Heijde, “The shape of the anterior and posterior surface of the aging human cornea,” Vision Res. 46(6-7), 993–1001 (2006).
[Crossref] [PubMed]

M. Dubbelman, G. L. van der Heijde, and H. A. Weeber, “The thickness of the aging human lens obtained from corrected Scheimpflug images,” Optom. Vis. Sci. 78(6), 411–416 (2001).
[Crossref] [PubMed]

M. Dubbelman and G. L. Van der Heijde, “The shape of the aging human lens: Curvature, equivalent refractive index and the lens paradox,” Vision Res. 41(14), 1867–1877 (2001).
[Crossref] [PubMed]

Dufault, P.

O. Pomerantzeff, M. Pankratov, G. J. Wang, and P. Dufault, “Wide-angle optical model of the eye,” Am. J. Optom. Physiol. Opt. 61(3), 166–176 (1984).
[Crossref] [PubMed]

Eva, P. R.

P. R. Eva, P. T. Pascoe, and D. G. Vaughan, “Refractive change in hyperglycaemia: hyperopia, not myopia,” Br. J. Ophthalmol. 66(8), 500–505 (1982).
[Crossref] [PubMed]

Evans, T.

T. Evans and W. F. Harris, “Dependence of the transference of a reduced eye on frequency of light,” S. Afr. Optom. 70, 149–155 (2011).

Fowler, C. W.

N. Drasdo and C. W. Fowler, “Non-linear projection of the retinal image in a wide-angle schematic eye,” Br. J. Ophthalmol. 58(8), 709–714 (1974).
[Crossref] [PubMed]

Goncharov, A. V.

González, L.

Harris, W. F.

T. Evans and W. F. Harris, “Dependence of the transference of a reduced eye on frequency of light,” S. Afr. Optom. 70, 149–155 (2011).

W. F. Harris, “Magnification, blur, and ray state at the retina for the general eye with and without a general optical instrument in front of it: 1. Distant Objects,” Optom. Vis. Sci. 78(12), 888–900 (2001).
[Crossref] [PubMed]

Hoshino, M.

B. Pierscionek, M. Bahrami, M. Hoshino, K. Uesugi, J. Regini, and N. Yagi, “The eye lens: age-related trends and individual variations in refractive index and shape parameters,” Oncotarget 6(31), 30532–30544 (2015).
[Crossref] [PubMed]

Jones, C. E.

C. E. Jones, D. A. Atchison, R. Meder, and J. M. Pope, “Refractive index distribution and optical properties of the isolated human lens measured using magnetic resonance imaging (MRI),” Vision Res. 45(18), 2352–2366 (2005).
[Crossref] [PubMed]

Kasthurirangan, S.

D. A. Atchison, E. L. Markwell, S. Kasthurirangan, J. M. Pope, G. Smith, and P. G. Swann, “Age-related changes in optical and biometric characteristics of emmetropic eyes,” J. Vis. 8(4), 29 (2008).
[Crossref] [PubMed]

Klein, B. E.

B. E. Klein, K. E. Lee, and R. Klein, “Refraction in adults with diabetes,” Arch. Ophthalmol. 129(1), 56–62 (2011).
[Crossref] [PubMed]

Klein, R.

B. E. Klein, K. E. Lee, and R. Klein, “Refraction in adults with diabetes,” Arch. Ophthalmol. 129(1), 56–62 (2011).
[Crossref] [PubMed]

Kooijman, A. C.

Lahm, D.

D. Lahm, L. K. Lee, and F. A. Bettelheim, “Age dependence of freezable and nonfreezable water content of normal human lenses,” Invest. Ophthalmol. Vis. Sci. 26(8), 1162–1165 (1985).
[PubMed]

Lee, K. E.

B. E. Klein, K. E. Lee, and R. Klein, “Refraction in adults with diabetes,” Arch. Ophthalmol. 129(1), 56–62 (2011).
[Crossref] [PubMed]

Lee, L. K.

D. Lahm, L. K. Lee, and F. A. Bettelheim, “Age dependence of freezable and nonfreezable water content of normal human lenses,” Invest. Ophthalmol. Vis. Sci. 26(8), 1162–1165 (1985).
[PubMed]

Lin, P. K.

S. F. Lin, P. K. Lin, F. L. Chang, and R. K. Tsai, “Transient hyperopia after intensive treatment of hyperglycemia in newly diagnosed diabetes,” Ophthalmologica 223(1), 68–71 (2009).
[Crossref] [PubMed]

Lin, S. F.

S. F. Lin, P. K. Lin, F. L. Chang, and R. K. Tsai, “Transient hyperopia after intensive treatment of hyperglycemia in newly diagnosed diabetes,” Ophthalmologica 223(1), 68–71 (2009).
[Crossref] [PubMed]

Liou, H. L.

Lopez-Gil, N.

Lotmar, W.

Markwell, E. L.

D. A. Atchison, E. L. Markwell, S. Kasthurirangan, J. M. Pope, G. Smith, and P. G. Swann, “Age-related changes in optical and biometric characteristics of emmetropic eyes,” J. Vis. 8(4), 29 (2008).
[Crossref] [PubMed]

Meder, R.

C. E. Jones, D. A. Atchison, R. Meder, and J. M. Pope, “Refractive index distribution and optical properties of the isolated human lens measured using magnetic resonance imaging (MRI),” Vision Res. 45(18), 2352–2366 (2005).
[Crossref] [PubMed]

Navarro, R.

Norrby, S.

S. Norrby, “The Dubbelman eye model analysed by ray tracing through aspheric surfaces,” Ophthalmic Physiol. Opt. 25(2), 153–161 (2005).
[Crossref] [PubMed]

Palos, F.

Pankratov, M.

O. Pomerantzeff, M. Pankratov, G. J. Wang, and P. Dufault, “Wide-angle optical model of the eye,” Am. J. Optom. Physiol. Opt. 61(3), 166–176 (1984).
[Crossref] [PubMed]

Pascoe, P. T.

P. R. Eva, P. T. Pascoe, and D. G. Vaughan, “Refractive change in hyperglycaemia: hyperopia, not myopia,” Br. J. Ophthalmol. 66(8), 500–505 (1982).
[Crossref] [PubMed]

Pierscionek, B.

B. Pierscionek, M. Bahrami, M. Hoshino, K. Uesugi, J. Regini, and N. Yagi, “The eye lens: age-related trends and individual variations in refractive index and shape parameters,” Oncotarget 6(31), 30532–30544 (2015).
[Crossref] [PubMed]

B. Pierscionek, “Presbyopia–effect of refractive index,” Clin. Exp. Optom. 73(1), 23–30 (1990).
[Crossref]

Pierscionek, B. K.

B. K. Pierscionek and J. W. Regini, “The gradient index lens of the eye: an opto-biological synchrony,” Prog. Retin. Eye Res. 31(4), 332–349 (2012).
[Crossref] [PubMed]

G. Smith and B. K. Pierscionek, “The optical structure of the lens and its contribution to the refractive status of the eye,” Ophthalmic Physiol. Opt. 18(1), 21–29 (1998).
[Crossref] [PubMed]

G. Smith, D. A. Atchison, and B. K. Pierscionek, “Modelling the ageing human eye,” J. Opt. Soc. Am. A 9(12), 2111–2117 (1992).
[Crossref]

Pomerantzeff, O.

O. Pomerantzeff, M. Pankratov, G. J. Wang, and P. Dufault, “Wide-angle optical model of the eye,” Am. J. Optom. Physiol. Opt. 61(3), 166–176 (1984).
[Crossref] [PubMed]

Pope, J. M.

D. A. Atchison, E. L. Markwell, S. Kasthurirangan, J. M. Pope, G. Smith, and P. G. Swann, “Age-related changes in optical and biometric characteristics of emmetropic eyes,” J. Vis. 8(4), 29 (2008).
[Crossref] [PubMed]

C. E. Jones, D. A. Atchison, R. Meder, and J. M. Pope, “Refractive index distribution and optical properties of the isolated human lens measured using magnetic resonance imaging (MRI),” Vision Res. 45(18), 2352–2366 (2005).
[Crossref] [PubMed]

Regini, J.

B. Pierscionek, M. Bahrami, M. Hoshino, K. Uesugi, J. Regini, and N. Yagi, “The eye lens: age-related trends and individual variations in refractive index and shape parameters,” Oncotarget 6(31), 30532–30544 (2015).
[Crossref] [PubMed]

Regini, J. W.

B. K. Pierscionek and J. W. Regini, “The gradient index lens of the eye: an opto-biological synchrony,” Prog. Retin. Eye Res. 31(4), 332–349 (2012).
[Crossref] [PubMed]

Santamaría, J.

Sheil, C. J.

Sicam, V. A.

M. Dubbelman, V. A. Sicam, and G. L. Van der Heijde, “The shape of the anterior and posterior surface of the aging human cornea,” Vision Res. 46(6-7), 993–1001 (2006).
[Crossref] [PubMed]

Smith, G.

D. A. Atchison, E. L. Markwell, S. Kasthurirangan, J. M. Pope, G. Smith, and P. G. Swann, “Age-related changes in optical and biometric characteristics of emmetropic eyes,” J. Vis. 8(4), 29 (2008).
[Crossref] [PubMed]

G. Smith and B. K. Pierscionek, “The optical structure of the lens and its contribution to the refractive status of the eye,” Ophthalmic Physiol. Opt. 18(1), 21–29 (1998).
[Crossref] [PubMed]

G. Smith, D. A. Atchison, and B. K. Pierscionek, “Modelling the ageing human eye,” J. Opt. Soc. Am. A 9(12), 2111–2117 (1992).
[Crossref]

Swann, P. G.

D. A. Atchison, E. L. Markwell, S. Kasthurirangan, J. M. Pope, G. Smith, and P. G. Swann, “Age-related changes in optical and biometric characteristics of emmetropic eyes,” J. Vis. 8(4), 29 (2008).
[Crossref] [PubMed]

Tsai, R. K.

S. F. Lin, P. K. Lin, F. L. Chang, and R. K. Tsai, “Transient hyperopia after intensive treatment of hyperglycemia in newly diagnosed diabetes,” Ophthalmologica 223(1), 68–71 (2009).
[Crossref] [PubMed]

Uesugi, K.

B. Pierscionek, M. Bahrami, M. Hoshino, K. Uesugi, J. Regini, and N. Yagi, “The eye lens: age-related trends and individual variations in refractive index and shape parameters,” Oncotarget 6(31), 30532–30544 (2015).
[Crossref] [PubMed]

Van der Heijde, G. L.

M. Dubbelman, V. A. Sicam, and G. L. Van der Heijde, “The shape of the anterior and posterior surface of the aging human cornea,” Vision Res. 46(6-7), 993–1001 (2006).
[Crossref] [PubMed]

M. Dubbelman, G. L. van der Heijde, and H. A. Weeber, “The thickness of the aging human lens obtained from corrected Scheimpflug images,” Optom. Vis. Sci. 78(6), 411–416 (2001).
[Crossref] [PubMed]

M. Dubbelman and G. L. Van der Heijde, “The shape of the aging human lens: Curvature, equivalent refractive index and the lens paradox,” Vision Res. 41(14), 1867–1877 (2001).
[Crossref] [PubMed]

Vaughan, D. G.

P. R. Eva, P. T. Pascoe, and D. G. Vaughan, “Refractive change in hyperglycaemia: hyperopia, not myopia,” Br. J. Ophthalmol. 66(8), 500–505 (1982).
[Crossref] [PubMed]

Wang, G. J.

O. Pomerantzeff, M. Pankratov, G. J. Wang, and P. Dufault, “Wide-angle optical model of the eye,” Am. J. Optom. Physiol. Opt. 61(3), 166–176 (1984).
[Crossref] [PubMed]

Weeber, H. A.

M. Dubbelman, G. L. van der Heijde, and H. A. Weeber, “The thickness of the aging human lens obtained from corrected Scheimpflug images,” Optom. Vis. Sci. 78(6), 411–416 (2001).
[Crossref] [PubMed]

Yagi, N.

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Figures (7)

Fig. 1
Fig. 1 Power of the eye model against the number of shells inside the lens (age 20 years).
Fig. 2
Fig. 2 Magnitudes of the radii of curvature of the internal shells of the lens against normalized distance d. Four different power laws (1, 3, 5, and 7) are presented. Parameters are based on an eye of age 20 years.
Fig. 3
Fig. 3 The index of refraction n i , position d inside the lens, and magnitude of the radii of curvature r i for anterior and posterior part of the i-th shell for an eye of age 20 years and using power law 5.
Fig. 4
Fig. 4 Lens model of (a) 20 year old, (b) 45 year old, and (c) 70 year old, with lens nucleus of index of refraction n max ( Age ) (Eq. (2) and power law 5. All measurements are in mm.
Fig. 5
Fig. 5 Power of the eye model as a function of age according to parameters given in Table 1 and n max =1.43. The red dashed line gives the power of the eye model with power law 1 and the black solid line to power law 5 and using n max ( Age ).
Fig. 6
Fig. 6 Radius of curvature of the nucleus r N as a function of age for different power laws. Radii of curvature of the anterior (rla - red dotted) and magnitude of the posterior (rlp - black dotted) lens surfaces are also given. The linear fits for r N are given. The R 2 values are 0.9949, 0.9962 and 0.9968 for power laws 3, 5 and 7, respectively. p<0.001 for all r N fits.
Fig. 7
Fig. 7 Radius of curvature of the nucleus as a function of age for five different nuclear maximum index values using power law 5. The linear fits for r N are given. Radii of curvature of the anterior (rla - red dotted) and magnitude of the posterior (rlp - black dotted) lens surfaces are also given. The R 2 values are 0.9951, 0,9957, 0.9962, 0.9965 and 0.9964 for n max values of 1,41, 1.42, 1.43, n max ( Age ) and 1.44, respectively. p<0.001 for all r N fits

Tables (1)

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Table 1 Data for the eye model.

Equations (9)

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n= n max +( n min n max ) d 2g
n max ( Age )=1.43850.0001Age.
M=( A B C D )= T PC L T AC R K2 T K R K1 ,
R=( 1 0 ( n 0 n 1 )/r 1 )
T=( 1 t/n 0 1 )
r i =a d p +b d p1 + r N
M 20 =( 0.0000 15.9008× 10 -3 m 62.8893D 0.8976 )
M 45 =( 0.0000 16.1410× 10 -3 m 61.9546D 0.9022 )
M 70 =( 0.0000 16.3908× 10 -3 m 61.0104D 0.9074 )

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