The relative importance of the various optical elements of the human eye are analyzed to determine which contribute most to the chromatic variance in total refractive power of the eye. The concept of differential dispersion, defined as the change in the difference in index of refraction across a refractive surface with change in wavelength, is used to provide a theoretical tool for this analysis. The theoretical treatment shows that almost all the chromatic effect will be caused by the air–tear interface. Calculations of model eyes are made that support this view. Four model eyes are examined, an emmetropic eye, a hyperopic eye, a myopic eye, and an emmetropic eye accommodating .
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Values of the Cauchy Coefficients A, B, C, and D Used to Calculate the Indices of Refraction of the Ocular Mediaa
Media (Index)
A
B
C
D
Air—1
1
0
0
0
Tear—2
1.321631
Cornea—3
1.361594
Aqueous—4
1.321631
Lens cortex—5
1.355
Lens nucleus—6
1.404
Vitreous—7
1.322357
To use these coefficients in Eq. (3), the values of the wavelengths of light are to be in units of nanometers. Coefficient A is unitless, as is the index of refraction. The units of the other coefficients are B ( squared), C ( to the fourth power), and D ( to the sixth power).
Table 2
Differences , , and in Cauchy Coefficient Values at the Refractive Surface Interfaces of the Human Eye Based on the Values of B, C, and D in Table 1a
Surface
Air–tear
6070.796
Anterior cornea
30154500
Posterior cornea
61.109
Anterior lens
450.422
95164480
Anterior cortex–nucleus
0.0
0.0
0.0
Posterior cortex–nucleus
0.0
0.0
0.0
Posterior lens
29327000
The units of the coefficient differences are ( squared), ( to the fourth power) and ( to the sixth power).
Table 3
Values of the Central Radii of Curvature, the Distances between Refractive Surfaces, and the Rate of Change of Surface Power with Wavelength of Light Change for the Refractive Surfaces of the Emmetropic Eyea
Surface
Radius of Curvature (mm)
Distance from Prior Surface (mm)
(D)
Retina
—
0
—
Posterior lens
17.7
Posterior cortex–nucleus
0.34
0
Anterior cortex–nucleus
11.53
2.2
0
Anterior lens
12.4
0.87
Posterior cornea
6.4
3.6
Anterior cornea
7.76
0.52
Air–tear
7.77
0.01
The rates of change of refractive powers are calculated for a wavelength of .
Table 4
Surface Power Values in Diopters for the Refractive Surfaces of the Emmetropic Eye Calculated for Wavelengths of , , and a
Wavelength (nm)
F (Air–Tear)
F (Anterior Cornea)
F (Posterior Cornea)
F (Anterior Lens)
F (Anterior Cortex – Nucleus)
F (Posterior Cortex – Nucleus)
F (Posterior Lens)
400
44.66
5.20
3.17
4.25
6.78
4.89
578
43.13
5.16
2.86
4.25
6.78
4.38
800
42.42
5.15
2.77
4.25
6.78
4.21
Total power change
2.24
0.06
0.40
0.00
0.00
0.68
The difference in refractive power between and for each refractive surface is also given (last line) to show the relative change in surface power of each refractive surface over the range of visible wavelengths of light. The relative changes in power values clearly show which surfaces will contribute the most to chromatic error of the eye. By a large amount, the air–tear surface is seen to have the dominant effect.
Table 5
Values of the Reduced Vergence as Light Progresses from the Retina to Exit from the Emmetropic Eye Are Given in Diopters for Each Refractive Surface at Wavelengths of , , and a
At each refracting surface the change in reduced vergence between and is given (last row).
Table 6
Values of the Central Radii of Curvature of and the Distances between the Refractive Surfaces of the Hyperopic Eye
Surface
Radius of Curvature (mm)
Distance from Prior Surface (mm)
Retina
—
0
Posterior lens
16.14
Posterior cortex–nucleus
0.34
Anterior corte–nucleus
11.53
2.2
Anterior lens
12.4
0.87
Posterior cornea
6.4
3.6
Anterior cornea
7.76
0.52
Air–tear
7.77
0.01
Table 7
Values of the Central Radii of Curvature of and the Distances between the Refractive Surfaces of the Myopic Eye
Surface
Radius of Curvature (mm)
Distance from Prior Surface (mm)
Retina
—
0
Posterior lens
19.48
Posterior cortex–nucleus
0.34
Anterior cortex–nucleus
11.53
2.2
Anterior lens
12.4
0.87
Posterior cornea
6.4
3.6
Anterior cornea
7.76
0.52
Air–tear
7.77
0.01
Table 8
Values of the Reduced Vergence in Diopters as Light Progresses from the Retina to Exit from the Hyperopic Eye Are Given for Each Refractive Surface at Wavelengths of , , and a
For each refracting surface the change in reduced vergence between and is given (last row).
Table 9
Values of the Reduced Vergence in Diopters as Light Progresses from the Retina to Exit from the Myopic Eye Are Given for Each Refractive Surface at Wavelengths of , , and a
For each refracting surface the change in reduced vergence between and is given (last row).
Table 10
Values of the Central Radii of Curvature of and the Distances between the Refractive Surfaces of the Emmetropic Eye Accommodating
Surface
Radius of Curvature (mm)
Distance from Prior Surface (mm)
Retina
—
0
Posterior lens
17.7
Posterior cortex–nucleus
0.34
Anterior cortex–nucleus
9.55
2.42
Anterior lens
10.42
0.87
Posterior cornea
6.4
3.38
Anterior cornea
7.76
0.52
Air–tear
7.77
0.01
Table 11
Values of the Reduced Vergence in Diopters as Light Progresses from the Retina to Exit from the Emmetropic Eye Accommodating Are Given for Each Refractive Surface at Wavelengths of , , and a
For each refracting surface the change in reduced vergence between and is given (last line).
Table 12
Differences in Power in Diopters of the Air–Tear Film Refractive Surface, the Calculated Total Refractive Error, and the Experimentally Measured Total Refractive Error between the Value Found at for the Emmetropic Eye and Nine Other Wavelengths in the Visual Spectral Rangea
Wavelength (nm)
Calculated Change in the Power of the Air–Tear Surface
Calculated Change in Refractive Error
Measured Change in Refractive Error
Difference in Refractive Change– Measured Minus Calculated
400
1.53
1.56
1.80
0.24
450
0.86
0.87
1.10
0.23
500
0.44
0.44
0.56
0.12
550
0.14
0.14
0.17
0.03
578
0
0
0
0
600
650
700
0.00
750
0.01
800
0.01
The difference between the calculated total refractive error and experimentally measured total refractive error (measured minus calculated) is also given (last column) at each wavelength. The difference values are found by subtracting the value at from that for the wavelength of interest.
Tables (12)
Table 1
Values of the Cauchy Coefficients A, B, C, and D Used to Calculate the Indices of Refraction of the Ocular Mediaa
Media (Index)
A
B
C
D
Air—1
1
0
0
0
Tear—2
1.321631
Cornea—3
1.361594
Aqueous—4
1.321631
Lens cortex—5
1.355
Lens nucleus—6
1.404
Vitreous—7
1.322357
To use these coefficients in Eq. (3), the values of the wavelengths of light are to be in units of nanometers. Coefficient A is unitless, as is the index of refraction. The units of the other coefficients are B ( squared), C ( to the fourth power), and D ( to the sixth power).
Table 2
Differences , , and in Cauchy Coefficient Values at the Refractive Surface Interfaces of the Human Eye Based on the Values of B, C, and D in Table 1a
Surface
Air–tear
6070.796
Anterior cornea
30154500
Posterior cornea
61.109
Anterior lens
450.422
95164480
Anterior cortex–nucleus
0.0
0.0
0.0
Posterior cortex–nucleus
0.0
0.0
0.0
Posterior lens
29327000
The units of the coefficient differences are ( squared), ( to the fourth power) and ( to the sixth power).
Table 3
Values of the Central Radii of Curvature, the Distances between Refractive Surfaces, and the Rate of Change of Surface Power with Wavelength of Light Change for the Refractive Surfaces of the Emmetropic Eyea
Surface
Radius of Curvature (mm)
Distance from Prior Surface (mm)
(D)
Retina
—
0
—
Posterior lens
17.7
Posterior cortex–nucleus
0.34
0
Anterior cortex–nucleus
11.53
2.2
0
Anterior lens
12.4
0.87
Posterior cornea
6.4
3.6
Anterior cornea
7.76
0.52
Air–tear
7.77
0.01
The rates of change of refractive powers are calculated for a wavelength of .
Table 4
Surface Power Values in Diopters for the Refractive Surfaces of the Emmetropic Eye Calculated for Wavelengths of , , and a
Wavelength (nm)
F (Air–Tear)
F (Anterior Cornea)
F (Posterior Cornea)
F (Anterior Lens)
F (Anterior Cortex – Nucleus)
F (Posterior Cortex – Nucleus)
F (Posterior Lens)
400
44.66
5.20
3.17
4.25
6.78
4.89
578
43.13
5.16
2.86
4.25
6.78
4.38
800
42.42
5.15
2.77
4.25
6.78
4.21
Total power change
2.24
0.06
0.40
0.00
0.00
0.68
The difference in refractive power between and for each refractive surface is also given (last line) to show the relative change in surface power of each refractive surface over the range of visible wavelengths of light. The relative changes in power values clearly show which surfaces will contribute the most to chromatic error of the eye. By a large amount, the air–tear surface is seen to have the dominant effect.
Table 5
Values of the Reduced Vergence as Light Progresses from the Retina to Exit from the Emmetropic Eye Are Given in Diopters for Each Refractive Surface at Wavelengths of , , and a
At each refracting surface the change in reduced vergence between and is given (last row).
Table 6
Values of the Central Radii of Curvature of and the Distances between the Refractive Surfaces of the Hyperopic Eye
Surface
Radius of Curvature (mm)
Distance from Prior Surface (mm)
Retina
—
0
Posterior lens
16.14
Posterior cortex–nucleus
0.34
Anterior corte–nucleus
11.53
2.2
Anterior lens
12.4
0.87
Posterior cornea
6.4
3.6
Anterior cornea
7.76
0.52
Air–tear
7.77
0.01
Table 7
Values of the Central Radii of Curvature of and the Distances between the Refractive Surfaces of the Myopic Eye
Surface
Radius of Curvature (mm)
Distance from Prior Surface (mm)
Retina
—
0
Posterior lens
19.48
Posterior cortex–nucleus
0.34
Anterior cortex–nucleus
11.53
2.2
Anterior lens
12.4
0.87
Posterior cornea
6.4
3.6
Anterior cornea
7.76
0.52
Air–tear
7.77
0.01
Table 8
Values of the Reduced Vergence in Diopters as Light Progresses from the Retina to Exit from the Hyperopic Eye Are Given for Each Refractive Surface at Wavelengths of , , and a
For each refracting surface the change in reduced vergence between and is given (last row).
Table 9
Values of the Reduced Vergence in Diopters as Light Progresses from the Retina to Exit from the Myopic Eye Are Given for Each Refractive Surface at Wavelengths of , , and a
For each refracting surface the change in reduced vergence between and is given (last row).
Table 10
Values of the Central Radii of Curvature of and the Distances between the Refractive Surfaces of the Emmetropic Eye Accommodating
Surface
Radius of Curvature (mm)
Distance from Prior Surface (mm)
Retina
—
0
Posterior lens
17.7
Posterior cortex–nucleus
0.34
Anterior cortex–nucleus
9.55
2.42
Anterior lens
10.42
0.87
Posterior cornea
6.4
3.38
Anterior cornea
7.76
0.52
Air–tear
7.77
0.01
Table 11
Values of the Reduced Vergence in Diopters as Light Progresses from the Retina to Exit from the Emmetropic Eye Accommodating Are Given for Each Refractive Surface at Wavelengths of , , and a
For each refracting surface the change in reduced vergence between and is given (last line).
Table 12
Differences in Power in Diopters of the Air–Tear Film Refractive Surface, the Calculated Total Refractive Error, and the Experimentally Measured Total Refractive Error between the Value Found at for the Emmetropic Eye and Nine Other Wavelengths in the Visual Spectral Rangea
Wavelength (nm)
Calculated Change in the Power of the Air–Tear Surface
Calculated Change in Refractive Error
Measured Change in Refractive Error
Difference in Refractive Change– Measured Minus Calculated
400
1.53
1.56
1.80
0.24
450
0.86
0.87
1.10
0.23
500
0.44
0.44
0.56
0.12
550
0.14
0.14
0.17
0.03
578
0
0
0
0
600
650
700
0.00
750
0.01
800
0.01
The difference between the calculated total refractive error and experimentally measured total refractive error (measured minus calculated) is also given (last column) at each wavelength. The difference values are found by subtracting the value at from that for the wavelength of interest.