Of the commonly used chromatic dispersion equations, only the Sellmeier and the Cauchy equations seem to be theoretically based. Cauchy’s equation is derived from the Sellmeier equation, is simpler to implement, and was found to give an excellent fit to published refractive-index data of the human eye. We used Cauchy’s equation to model the chromatic difference in refraction of the Gullstrand number 1 schematic eye with a gradient-index lens. To estimate the dispersion at different refractive-index levels within the lens, a single dispersion equation at one nominal refractive index was linearly scaled. This scaling was justified after exploring the effect of mean refractive index on dispersion by using Sellmeier’s equation and finding that a dispersion equation for one wavelength is just a linearly scaled version of the dispersion equation at any other wavlength. Because Cauchy’s equation is theoretically based and gives excellent fit to data in the visible spectrum, it can be used to extrapolate results into the near infrared with confidence.

Alexander V. Goncharov and Chris Dainty J. Opt. Soc. Am. A 24(8) 2157-2174 (2007)

References

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Coefficients in the Cauchy equation (3a).
Maximum difference between the data points and the value predicted by the equations.
Wavelength corresponding to max. diff.

Table 5

Coefficients for the Cauchy Equation for Each Ocular Medium, Taken from the Combined Data of Le Grand14
and Navarro et al.15a

A

B

C

D

n(555)

V
value

W
value

Cornea

1.361594

6.009687 × 10^{3}

-6.760760 × 10^{8}

5.908450 × 10^{13}

1.376000

55.48

0.43

Aqueous

1.321631

6.070796 × 10^{3}

-7.062305 × 10^{8}

6.147861 × 10^{13}

1.336000

50.37

0.43

Lens

High

1.389248

6.521218 × 10^{3}

-6.110661 × 10^{8}

5.908191 × 10^{13}

1.406000

47.32

0.40

Low

1.369486

6.428455 × 10^{3}

-6.023738 × 10^{8}

5.824149 × 10^{13}

1.386000

45.64

0.40

Vitreous

1.322357

5.560240 × 10^{3}

-5.817391 × 10^{8}

5.036810 × 10^{13}

1.336000

51.30

0.43

For the gradient-index lens, only those for the maximum and minimum values are given. For the lens, either the high- or the low-index values above can be scaled with Eq. (10).

Tables (5)

Table 1

Le Grand’s14
Chromatic Dispersion Data That Are Substituted into Eq. (3c)

${n}_{\infty}$

K

${\mathit{\lambda}}_{o}(\mathrm{nm})$

V

Cornea

1.3610

7.4147

130.0

56

Aqueous

1.3221

7.0096

130.0

53

Lens

1.3999

9.2492

130.0

50

Vitreous

1.3208

6.9806

130.0

53

Table 2

Values of ${n}^{**},{n}_{\mathbf{F}},{n}_{\mathbf{C}}$
and ${n}^{*}$
in Eq. (7a) for the Navarro et al15
Eye and Their Corresponding Wavelengths

${n}^{**}$
(0.365 µm)

${n}_{\mathrm{F}}$
(0.4861 µm)

${n}_{\mathrm{C}}$
(0.6563 µm)

${n}^{*}$
(1.014 µm)

Cornea

1.3975

1.3807

1.37405

1.3668

Aqueous

1.3593

1.3422

1.3354

1.3278

Lens

1.4492

1.42625

1.4175

1.4097

Aqueous

1.3565

1.3407

1.3341

1.3273

Table 3

Values of ${A}_{0},{A}_{1},$P
, and R
Used in Eq. (7b) To Determine ${a}_{1},{a}_{2},{a}_{3},$
and ${a}_{4}$
for the Navarro et al.15
Eye

Coefficient

A_{0}

A_{1}

P

R

a_{1}

0.66147196

-0.40352796

-0.28046790

0.03385979

a_{2}

-4.20146383

2.73508956

1.50543784

-0.11593235

a_{3}

6.29834237

-4.69409935

-1.57508650

0.10293038

a_{4}

-1.75835059

2.36253794

0.35011657

-0.02085782

Table 4

Cauchy’s Equations Fitted to Various Sources of Chromatic Dispersion Dataa

Coefficients in the Cauchy equation (3a).
Maximum difference between the data points and the value predicted by the equations.
Wavelength corresponding to max. diff.

Table 5

Coefficients for the Cauchy Equation for Each Ocular Medium, Taken from the Combined Data of Le Grand14
and Navarro et al.15a

A

B

C

D

n(555)

V
value

W
value

Cornea

1.361594

6.009687 × 10^{3}

-6.760760 × 10^{8}

5.908450 × 10^{13}

1.376000

55.48

0.43

Aqueous

1.321631

6.070796 × 10^{3}

-7.062305 × 10^{8}

6.147861 × 10^{13}

1.336000

50.37

0.43

Lens

High

1.389248

6.521218 × 10^{3}

-6.110661 × 10^{8}

5.908191 × 10^{13}

1.406000

47.32

0.40

Low

1.369486

6.428455 × 10^{3}

-6.023738 × 10^{8}

5.824149 × 10^{13}

1.386000

45.64

0.40

Vitreous

1.322357

5.560240 × 10^{3}

-5.817391 × 10^{8}

5.036810 × 10^{13}

1.336000

51.30

0.43

For the gradient-index lens, only those for the maximum and minimum values are given. For the lens, either the high- or the low-index values above can be scaled with Eq. (10).