Abstract

Attempts to correct the chromatic difference of focus of the human eye will introduce unwanted chromatic parallax if the eye is misaligned with the optical axis of the achromatizing system. Using geometrical optics, we show that the amount of parallax is approximately proportional to the amount of misalignment of the eye, with the constant of proportionality equal to the eye’s chromatic difference of refractive error. This prediction was confirmed by the experimental determination of chromatic parallax for two commercially available achromatizing lenses. On the basis of these results, we calculated that anticipated improvements in the polychromatic modulation transfer function of the eye offered by achromatizing lenses will be canceled by ~0.4 mm of the misalignment between the lens and the eye. Our prediction that further misalignment would severely reduce image quality of the achromatized eye was verified by psychophysical measurements of contrast sensitivity.

© 1991 Optical Society of America

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References

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  1. G. Wald, D. R. Griffin, “The change in refractive power of the human eye in dim and bright light,”J. Opt. Soc. Am. 37, 321–336 (1947).
    [CrossRef] [PubMed]
  2. R. E. Bedford, G. Wyszecki, “Axial chromatic aberration of the human eye,”J. Opt. Soc. Am. 47, 564–565 (1957).
    [CrossRef] [PubMed]
  3. P. A. Howarth, A. Bradley, “The longitudinal chromatic aberration of the human eye, and its correction,” Vision Res. 26, 361–366 (1986).
    [CrossRef] [PubMed]
  4. H. von Helmholtz, Treatise on Physiological Optics, 3rd ed., J. P. C. Southall, ed. (Optical Society of America, Washington, D.C., 1924), Vol. 3.
  5. A. C. S. van Heel, “Correcting the spherical and chromatic aberrations of the eye,”J. Opt. Soc. Am. 36, 237–239 (1946).
    [PubMed]
  6. H. Hartridge, “The visual perception of fine detail,” Philos. Trans. R. Soc. London Ser. B 232, 519–671 (1947).
    [CrossRef]
  7. L. C. Thomson, W. D. Wright, “The colour sensitivity of the retina within the central fovea of man,”J. Physiol. 105, 316–331 (1947).
    [PubMed]
  8. G. A. Fry, “Visibility of color contrast borders,” Am. J. Optom. Arch. Am. Acad. Optom. 49, 401–406 (1972).
    [CrossRef] [PubMed]
  9. I. Powell, “Lenses for correcting chromatic aberration of the eye,” Appl. Opt. 20, 4152–4155 (1981).
    [CrossRef] [PubMed]
  10. A. L. Lewis, M. Katz, C. Oehrlein, “A modified achromatizing lens,” Am. J. Optom. Physiol. Opt. 59, 909–911 (1982).
    [CrossRef] [PubMed]
  11. F. W. Campbell, “The depth-of-field of the human eye,” Opt. Acta 4, 157–164 (1957).
    [CrossRef]
  12. A. G. Bennett, J. Tucker, “Correspondence: chromatic aberration of the eye between 200 and 2000 nm,” Brit. J. Physiol. Opt. 30, 132–135 (1975).
  13. H. Hartridge, Recent Advances in the Physiology of Vision, 1st ed. (Churchill, London, 1950).
  14. F. W. Campbell, R. W. Gubisch, “The effect of chromatic aberration on visual acuity,”J. Physiol. 192, 345–358 (1967).
    [PubMed]
  15. D. G. Green, F. W. Campbell, “Effects of focus on the visual response to a sinusoidally modulated spatial stimulus,”J. Opt. Soc. Am. 55, 1154–1157 (1965).
    [CrossRef]
  16. A. van Meeteren, C. J. W. Dunnewold, “Image quality of the human eye for eccentric entrance pupil,” Vision Res. 23, 573–579 (1983).
    [CrossRef]
  17. L. N. Thibos, “Calculation of the influence of lateral chromatic aberration on image quality across the visual field,” J. Opt. Soc. Am. A 4, 1673–1680 (1987).
    [CrossRef] [PubMed]
  18. L. N. Thibos, A. Bradley, D. L. Still, X. Zhang, P. A. Howarth, “Theory and measurement of ocular chromatic aberration,” Vision Res. 30, 33–49 (1990).
    [CrossRef] [PubMed]
  19. T. W. Butler, L. A. Riggs, “Color difference scaled by chromatic modulation sensitivity functions,” Vision Res. 18, 1407–1416 (1978).
    [CrossRef]
  20. K. T. Mullen, “The contrast sensitivity of human colour vision to red–green and blue–yellow chromatic gratings,”J. Physiol. 359, 381–400 (1985).
  21. T. Troscianko, J. Harris, “Phase discrimination in chromatic compound gratings,” Vision Res. 28, 1041–1049 (1988).
    [CrossRef] [PubMed]
  22. Use of nodal ray instead of chief ray to determine image location is appropriate here because all wavelengths are simultaneously in focus on the retina when the eye is achromatized.
  23. A. Ivanoff, Les Aberrations de l’Oeil (Editions de la Revue D’Optique Theorique et Instrumentale, Paris, 1953).
  24. J. Guild, “Chromatic parallax and its influence on optical measurements,” Proc. Phys. Soc. 29, 311–338 (1917).
  25. Y. Le Grand, in Form and Space Vision, G. G. Heath, M. Millodot, eds. (Indiana U. Press, Bloomington, Ind., 1967).
  26. P. Simonet, M. C. W. Campbell, “The optical transverse chromatic aberration on the fovea of the human eye,” Vision Res. 30, 187–206 (1990).
    [CrossRef] [PubMed]
  27. C. F. Prentice, “The relation of the prism-dioptry to the lens-dioptry,” in Ophthalmic Lenses; Dioptric Formulae for Combined Cylindrical Lenses. The Prism-Dioptry and Other Optical Papers (Keystone, Philadelphia, 1907), pp. 115–124.
  28. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, San Francisco, 1968).
  29. H. H. Hopkins, “The frequency response of a defocused optical system,” Proc. R. Soc. London Ser. A 231, 91–103 (1955).
    [CrossRef]
  30. A. G. Bennett, R. B. Rabbetts, Clinical Visual Optics (Butterworth, London, 1984).

1990 (2)

L. N. Thibos, A. Bradley, D. L. Still, X. Zhang, P. A. Howarth, “Theory and measurement of ocular chromatic aberration,” Vision Res. 30, 33–49 (1990).
[CrossRef] [PubMed]

P. Simonet, M. C. W. Campbell, “The optical transverse chromatic aberration on the fovea of the human eye,” Vision Res. 30, 187–206 (1990).
[CrossRef] [PubMed]

1988 (1)

T. Troscianko, J. Harris, “Phase discrimination in chromatic compound gratings,” Vision Res. 28, 1041–1049 (1988).
[CrossRef] [PubMed]

1987 (1)

1986 (1)

P. A. Howarth, A. Bradley, “The longitudinal chromatic aberration of the human eye, and its correction,” Vision Res. 26, 361–366 (1986).
[CrossRef] [PubMed]

1985 (1)

K. T. Mullen, “The contrast sensitivity of human colour vision to red–green and blue–yellow chromatic gratings,”J. Physiol. 359, 381–400 (1985).

1983 (1)

A. van Meeteren, C. J. W. Dunnewold, “Image quality of the human eye for eccentric entrance pupil,” Vision Res. 23, 573–579 (1983).
[CrossRef]

1982 (1)

A. L. Lewis, M. Katz, C. Oehrlein, “A modified achromatizing lens,” Am. J. Optom. Physiol. Opt. 59, 909–911 (1982).
[CrossRef] [PubMed]

1981 (1)

1978 (1)

T. W. Butler, L. A. Riggs, “Color difference scaled by chromatic modulation sensitivity functions,” Vision Res. 18, 1407–1416 (1978).
[CrossRef]

1975 (1)

A. G. Bennett, J. Tucker, “Correspondence: chromatic aberration of the eye between 200 and 2000 nm,” Brit. J. Physiol. Opt. 30, 132–135 (1975).

1972 (1)

G. A. Fry, “Visibility of color contrast borders,” Am. J. Optom. Arch. Am. Acad. Optom. 49, 401–406 (1972).
[CrossRef] [PubMed]

1967 (1)

F. W. Campbell, R. W. Gubisch, “The effect of chromatic aberration on visual acuity,”J. Physiol. 192, 345–358 (1967).
[PubMed]

1965 (1)

1957 (2)

1955 (1)

H. H. Hopkins, “The frequency response of a defocused optical system,” Proc. R. Soc. London Ser. A 231, 91–103 (1955).
[CrossRef]

1947 (3)

G. Wald, D. R. Griffin, “The change in refractive power of the human eye in dim and bright light,”J. Opt. Soc. Am. 37, 321–336 (1947).
[CrossRef] [PubMed]

H. Hartridge, “The visual perception of fine detail,” Philos. Trans. R. Soc. London Ser. B 232, 519–671 (1947).
[CrossRef]

L. C. Thomson, W. D. Wright, “The colour sensitivity of the retina within the central fovea of man,”J. Physiol. 105, 316–331 (1947).
[PubMed]

1946 (1)

1917 (1)

J. Guild, “Chromatic parallax and its influence on optical measurements,” Proc. Phys. Soc. 29, 311–338 (1917).

Bedford, R. E.

Bennett, A. G.

A. G. Bennett, J. Tucker, “Correspondence: chromatic aberration of the eye between 200 and 2000 nm,” Brit. J. Physiol. Opt. 30, 132–135 (1975).

A. G. Bennett, R. B. Rabbetts, Clinical Visual Optics (Butterworth, London, 1984).

Bradley, A.

L. N. Thibos, A. Bradley, D. L. Still, X. Zhang, P. A. Howarth, “Theory and measurement of ocular chromatic aberration,” Vision Res. 30, 33–49 (1990).
[CrossRef] [PubMed]

P. A. Howarth, A. Bradley, “The longitudinal chromatic aberration of the human eye, and its correction,” Vision Res. 26, 361–366 (1986).
[CrossRef] [PubMed]

Butler, T. W.

T. W. Butler, L. A. Riggs, “Color difference scaled by chromatic modulation sensitivity functions,” Vision Res. 18, 1407–1416 (1978).
[CrossRef]

Campbell, F. W.

F. W. Campbell, R. W. Gubisch, “The effect of chromatic aberration on visual acuity,”J. Physiol. 192, 345–358 (1967).
[PubMed]

D. G. Green, F. W. Campbell, “Effects of focus on the visual response to a sinusoidally modulated spatial stimulus,”J. Opt. Soc. Am. 55, 1154–1157 (1965).
[CrossRef]

F. W. Campbell, “The depth-of-field of the human eye,” Opt. Acta 4, 157–164 (1957).
[CrossRef]

Campbell, M. C. W.

P. Simonet, M. C. W. Campbell, “The optical transverse chromatic aberration on the fovea of the human eye,” Vision Res. 30, 187–206 (1990).
[CrossRef] [PubMed]

Dunnewold, C. J. W.

A. van Meeteren, C. J. W. Dunnewold, “Image quality of the human eye for eccentric entrance pupil,” Vision Res. 23, 573–579 (1983).
[CrossRef]

Fry, G. A.

G. A. Fry, “Visibility of color contrast borders,” Am. J. Optom. Arch. Am. Acad. Optom. 49, 401–406 (1972).
[CrossRef] [PubMed]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, San Francisco, 1968).

Green, D. G.

Griffin, D. R.

Gubisch, R. W.

F. W. Campbell, R. W. Gubisch, “The effect of chromatic aberration on visual acuity,”J. Physiol. 192, 345–358 (1967).
[PubMed]

Guild, J.

J. Guild, “Chromatic parallax and its influence on optical measurements,” Proc. Phys. Soc. 29, 311–338 (1917).

Harris, J.

T. Troscianko, J. Harris, “Phase discrimination in chromatic compound gratings,” Vision Res. 28, 1041–1049 (1988).
[CrossRef] [PubMed]

Hartridge, H.

H. Hartridge, “The visual perception of fine detail,” Philos. Trans. R. Soc. London Ser. B 232, 519–671 (1947).
[CrossRef]

H. Hartridge, Recent Advances in the Physiology of Vision, 1st ed. (Churchill, London, 1950).

Hopkins, H. H.

H. H. Hopkins, “The frequency response of a defocused optical system,” Proc. R. Soc. London Ser. A 231, 91–103 (1955).
[CrossRef]

Howarth, P. A.

L. N. Thibos, A. Bradley, D. L. Still, X. Zhang, P. A. Howarth, “Theory and measurement of ocular chromatic aberration,” Vision Res. 30, 33–49 (1990).
[CrossRef] [PubMed]

P. A. Howarth, A. Bradley, “The longitudinal chromatic aberration of the human eye, and its correction,” Vision Res. 26, 361–366 (1986).
[CrossRef] [PubMed]

Ivanoff, A.

A. Ivanoff, Les Aberrations de l’Oeil (Editions de la Revue D’Optique Theorique et Instrumentale, Paris, 1953).

Katz, M.

A. L. Lewis, M. Katz, C. Oehrlein, “A modified achromatizing lens,” Am. J. Optom. Physiol. Opt. 59, 909–911 (1982).
[CrossRef] [PubMed]

Le Grand, Y.

Y. Le Grand, in Form and Space Vision, G. G. Heath, M. Millodot, eds. (Indiana U. Press, Bloomington, Ind., 1967).

Lewis, A. L.

A. L. Lewis, M. Katz, C. Oehrlein, “A modified achromatizing lens,” Am. J. Optom. Physiol. Opt. 59, 909–911 (1982).
[CrossRef] [PubMed]

Mullen, K. T.

K. T. Mullen, “The contrast sensitivity of human colour vision to red–green and blue–yellow chromatic gratings,”J. Physiol. 359, 381–400 (1985).

Oehrlein, C.

A. L. Lewis, M. Katz, C. Oehrlein, “A modified achromatizing lens,” Am. J. Optom. Physiol. Opt. 59, 909–911 (1982).
[CrossRef] [PubMed]

Powell, I.

Prentice, C. F.

C. F. Prentice, “The relation of the prism-dioptry to the lens-dioptry,” in Ophthalmic Lenses; Dioptric Formulae for Combined Cylindrical Lenses. The Prism-Dioptry and Other Optical Papers (Keystone, Philadelphia, 1907), pp. 115–124.

Rabbetts, R. B.

A. G. Bennett, R. B. Rabbetts, Clinical Visual Optics (Butterworth, London, 1984).

Riggs, L. A.

T. W. Butler, L. A. Riggs, “Color difference scaled by chromatic modulation sensitivity functions,” Vision Res. 18, 1407–1416 (1978).
[CrossRef]

Simonet, P.

P. Simonet, M. C. W. Campbell, “The optical transverse chromatic aberration on the fovea of the human eye,” Vision Res. 30, 187–206 (1990).
[CrossRef] [PubMed]

Still, D. L.

L. N. Thibos, A. Bradley, D. L. Still, X. Zhang, P. A. Howarth, “Theory and measurement of ocular chromatic aberration,” Vision Res. 30, 33–49 (1990).
[CrossRef] [PubMed]

Thibos, L. N.

L. N. Thibos, A. Bradley, D. L. Still, X. Zhang, P. A. Howarth, “Theory and measurement of ocular chromatic aberration,” Vision Res. 30, 33–49 (1990).
[CrossRef] [PubMed]

L. N. Thibos, “Calculation of the influence of lateral chromatic aberration on image quality across the visual field,” J. Opt. Soc. Am. A 4, 1673–1680 (1987).
[CrossRef] [PubMed]

Thomson, L. C.

L. C. Thomson, W. D. Wright, “The colour sensitivity of the retina within the central fovea of man,”J. Physiol. 105, 316–331 (1947).
[PubMed]

Troscianko, T.

T. Troscianko, J. Harris, “Phase discrimination in chromatic compound gratings,” Vision Res. 28, 1041–1049 (1988).
[CrossRef] [PubMed]

Tucker, J.

A. G. Bennett, J. Tucker, “Correspondence: chromatic aberration of the eye between 200 and 2000 nm,” Brit. J. Physiol. Opt. 30, 132–135 (1975).

van Heel, A. C. S.

van Meeteren, A.

A. van Meeteren, C. J. W. Dunnewold, “Image quality of the human eye for eccentric entrance pupil,” Vision Res. 23, 573–579 (1983).
[CrossRef]

von Helmholtz, H.

H. von Helmholtz, Treatise on Physiological Optics, 3rd ed., J. P. C. Southall, ed. (Optical Society of America, Washington, D.C., 1924), Vol. 3.

Wald, G.

Wright, W. D.

L. C. Thomson, W. D. Wright, “The colour sensitivity of the retina within the central fovea of man,”J. Physiol. 105, 316–331 (1947).
[PubMed]

Wyszecki, G.

Zhang, X.

L. N. Thibos, A. Bradley, D. L. Still, X. Zhang, P. A. Howarth, “Theory and measurement of ocular chromatic aberration,” Vision Res. 30, 33–49 (1990).
[CrossRef] [PubMed]

Am. J. Optom. Arch. Am. Acad. Optom. (1)

G. A. Fry, “Visibility of color contrast borders,” Am. J. Optom. Arch. Am. Acad. Optom. 49, 401–406 (1972).
[CrossRef] [PubMed]

Am. J. Optom. Physiol. Opt. (1)

A. L. Lewis, M. Katz, C. Oehrlein, “A modified achromatizing lens,” Am. J. Optom. Physiol. Opt. 59, 909–911 (1982).
[CrossRef] [PubMed]

Appl. Opt. (1)

Brit. J. Physiol. Opt. (1)

A. G. Bennett, J. Tucker, “Correspondence: chromatic aberration of the eye between 200 and 2000 nm,” Brit. J. Physiol. Opt. 30, 132–135 (1975).

J. Opt. Soc. Am. (4)

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

J. Physiol. (3)

K. T. Mullen, “The contrast sensitivity of human colour vision to red–green and blue–yellow chromatic gratings,”J. Physiol. 359, 381–400 (1985).

L. C. Thomson, W. D. Wright, “The colour sensitivity of the retina within the central fovea of man,”J. Physiol. 105, 316–331 (1947).
[PubMed]

F. W. Campbell, R. W. Gubisch, “The effect of chromatic aberration on visual acuity,”J. Physiol. 192, 345–358 (1967).
[PubMed]

Opt. Acta (1)

F. W. Campbell, “The depth-of-field of the human eye,” Opt. Acta 4, 157–164 (1957).
[CrossRef]

Philos. Trans. R. Soc. London Ser. B (1)

H. Hartridge, “The visual perception of fine detail,” Philos. Trans. R. Soc. London Ser. B 232, 519–671 (1947).
[CrossRef]

Proc. Phys. Soc. (1)

J. Guild, “Chromatic parallax and its influence on optical measurements,” Proc. Phys. Soc. 29, 311–338 (1917).

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

H. H. Hopkins, “The frequency response of a defocused optical system,” Proc. R. Soc. London Ser. A 231, 91–103 (1955).
[CrossRef]

Vision Res. (6)

P. Simonet, M. C. W. Campbell, “The optical transverse chromatic aberration on the fovea of the human eye,” Vision Res. 30, 187–206 (1990).
[CrossRef] [PubMed]

L. N. Thibos, A. Bradley, D. L. Still, X. Zhang, P. A. Howarth, “Theory and measurement of ocular chromatic aberration,” Vision Res. 30, 33–49 (1990).
[CrossRef] [PubMed]

T. W. Butler, L. A. Riggs, “Color difference scaled by chromatic modulation sensitivity functions,” Vision Res. 18, 1407–1416 (1978).
[CrossRef]

T. Troscianko, J. Harris, “Phase discrimination in chromatic compound gratings,” Vision Res. 28, 1041–1049 (1988).
[CrossRef] [PubMed]

P. A. Howarth, A. Bradley, “The longitudinal chromatic aberration of the human eye, and its correction,” Vision Res. 26, 361–366 (1986).
[CrossRef] [PubMed]

A. van Meeteren, C. J. W. Dunnewold, “Image quality of the human eye for eccentric entrance pupil,” Vision Res. 23, 573–579 (1983).
[CrossRef]

Other (8)

H. Hartridge, Recent Advances in the Physiology of Vision, 1st ed. (Churchill, London, 1950).

H. von Helmholtz, Treatise on Physiological Optics, 3rd ed., J. P. C. Southall, ed. (Optical Society of America, Washington, D.C., 1924), Vol. 3.

Use of nodal ray instead of chief ray to determine image location is appropriate here because all wavelengths are simultaneously in focus on the retina when the eye is achromatized.

A. Ivanoff, Les Aberrations de l’Oeil (Editions de la Revue D’Optique Theorique et Instrumentale, Paris, 1953).

Y. Le Grand, in Form and Space Vision, G. G. Heath, M. Millodot, eds. (Indiana U. Press, Bloomington, Ind., 1967).

C. F. Prentice, “The relation of the prism-dioptry to the lens-dioptry,” in Ophthalmic Lenses; Dioptric Formulae for Combined Cylindrical Lenses. The Prism-Dioptry and Other Optical Papers (Keystone, Philadelphia, 1907), pp. 115–124.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, San Francisco, 1968).

A. G. Bennett, R. B. Rabbetts, Clinical Visual Optics (Butterworth, London, 1984).

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Figures (6)

Fig. 1
Fig. 1

Achromatizing and chromatic parallax. (a) Ideal achromatizing case. The achromatizing lens is treated as a thin lens. l0 is the object distance (to the lens principal point or nodal point); N is the nodal point of the eye; points A and B are long- and short-wavelength images, respectively, formed by the achromatizing lens; la′ and lb′ are the image distances for A and B, respectively; z is the distance between the lens and the nodal point of the eye. (b) Chromatic parallax caused by lateral head movements. The lens and the object are removed for clarity, ye is the amount of head movement, and θ is the angular amount of chromatic parallax. (c) Chromatic parallax caused by lens movements. yl is the amount of lens movement.

Fig. 2
Fig. 2

Lens power plotted as a function of wavelength for the Powell lens (triangles) and the Lewis lens (squares). Also plotted is the chromatic difference in refractive error of human eyes (circles). Curves are best-fitting polynomials. Standard deviation of the lens power data are all smaller than the symbols.

Fig. 3
Fig. 3

Chromatic parallax measured with two achromatizing lenses. (a) External chromatic parallax determined with a Powell lens for subjects AB (open triangles) and XZ (filled triangles). The solid line shows model predictions for the Powell lens. (b) External chromatic parallax determined with a Lewis lens for subjects AB (open squares) and XZ (filled squares). The solid line shows model predictions for the Lewis lens. Standard deviations of the data are all smaller than the symbols.

Fig. 4
Fig. 4

Comparison of chromatic parallax created by moving the head (triangles) or the Powell lens (circles) on subject XZ. The solid line shows model predictions. Standard deviations of the data are all smaller than the symbols.

Fig. 5
Fig. 5

Predicted MTF’s for an achromatized model eye with different amounts of lens–eye misalignment (solid curves). MTF’s were calculated for a 2.5-mm pupil diameter and 2798-K tungsten light sources. Values near each curve indicate the amount of lens shift from the chromatic axis of the eye. For comparison the MTF without the achromatizing lens (i.e., longitudinal chromatic aberration uncorrected) is shown by the dashed curve.

Fig. 6
Fig. 6

Effects of chromatic parallax on grating contrast sensitivity. (a) Contrast sensitivity measured for subject AB with the achromatizing lens aligned on the achromatic axis (open triangles) and that with the lens misaligned by 2 mm (filled triangles). (b) Contrast-sensitivity ratios compare with predicted MTF ratios. Symbols are CSF ratios (misaligned 2 mm/aligned) for three subjects. The curve is MTF ratios (with parallax/without parallax). MTF’s were calculated for a 5-mm pupil diameter and P4 phosphor.

Equations (6)

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

θ = tan - 1 ( y e B C ) - tan - 1 ( y e A C ) = tan - 1 ( y e z - l b ) - tan - 1 ( y e z - l a ) ,
θ = y e ( 1 - l b - 1 - l a ) = y e ( F a - F b ) ,
F l = 1 l 1 - 1 l 2 ,
F λ = - 13.729 + ( 44.309 × 10 - 3 λ ) - ( 34.441 × 10 - 6 λ 2 )
F λ = - 12.807 + ( 33.246 × 10 - 3 λ ) - ( 29.98 × 10 - 6 λ 2 )
R x λ = - 11.822 + ( 35.726 × 10 - 3 λ ) - ( 26.308 × 10 - 6 λ 2 )

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