Howard S. Coleman, Madeline F. Coleman, David L. Fridge, and Samuel W. Harding, "The Coefficient of Specific Resolution of the Human Eye for Foucault Test Objects Viewed through Circular Apertures*," J. Opt. Soc. Am. 39, 766-770 (1949)

A study has been made of the resolving power of the human eye for Foucault test objects of different inherent contrast viewed through circular artificial pupil stops ranging from 0.30 of a millimeter to 7.0 millimeters in diameter. Two methods were used to make the resolution measurements. The first consisted of viewing the test objects through circular apertures in 10-mil blackened brass shim stock disks placed about 3 millimeters from the observer’s eye. Provision was made for illuminating the test objects both with tungsten lamps and with mercury arcs. The second method involved the use of test objects viewed in collimated light by means of a telescopic system for which the exit pupil could be varied from 0.40 of a millimeter to 2.0 millimeters. Both methods used equipment embodying the basic principles of the K.D.C. apparatus. The study included resolution measurements for rectangular and square Foucault test objects consisting of straight parallel black and white bands having an inherent contrast of approximately 94 percent. The coefficient of specific resolution was computed for test objects having lower values of inherent contrast. The resolution measurements, totaling over 100,000, were made by thirty-two different observers ranging from 18 to 76 years in age. The coefficient of specific resolution was found to be practically independent of age and observer for artificial pupil stops less than 0.75 of a millimeter in diameter. This suggests an effective consistency of the contrast threshold of the human retina as far as resolution measurements are concerned and indicates that the diffraction of light at the pupil stop is the primary factor controlling resolution. For artificial pupil stops of larger diameter, the coefficient of specific resolution was found to depend upon the individual observer but only with a maximum variation among the observers used of 30 percent. The composition of the test object was found to influence the coefficient of specific resolution to a certain extent with an indication that the best resolution was obtainable with test objects having the greatest number of elements.

Table I Minimum angle of resolution for a Foucault test object of a 94 percent inherent contrast vs. the diameter of an artificial pupil stop placed 3 mm from the pupil of the human eye.

Table II Minimum angle of resolution of a Foucault test object of 94 percent inherent contrast as a function of the exit pupil of a high grade telescopic system.

Table IV Computed values of the coefficient of specific resolution of Focault test objects of different inherent contrast viewed through an artificial pupil stop 0.4 of a mm in diameter.

Table V Computed values of the coefficient of specific resolution of Foucault test objects of different inherent contrast viewed through an artificial pupil stop 0.6 of a mm in diameter.

Table VI Computed values of the coefficient of specific resolution of Foucault test objects of different inherent contrast viewed through an artificial pupil stop 0.7 of a mm in diameter.

Table VII Computed values of the coefficient of specific resolution of Foucault test objects of different inherent contrast viewed through an artificial pupil stop 0.8 of a mm in diameter.

Table VIII Computed values of the coefficient of specific resolution of Foucault test objects of different inherent contrast viewed through an artificial pupil stop 1.0 mm in diameter.

Table IX Computed values of the coefficient of specific resolution of Foucault test objects of different inherent contrast viewed through an artificial pupil stop 1.5 mm in diameter.

Table X Computed values of the coefficient of specific resolution of Foucault test objects of different inherent contrast viewed through an artificial pupil stop 2.0 mm in diameter.

Minimum angle of resolution for a Foucault test object of a 94 percent inherent contrast vs. the diameter of an artificial pupil stop placed 3 mm from the pupil of the human eye.

Pupil stop(millimeters)

Minimum angle of resolution (radians ×10^{5}) for five observers

No. 1

No. 2

No. 3

No. 4

No. 5

0.41

144

133

133

137

133

0.46

129

122

125

125

125

0.76

81

76

80

79

77

1.02

67

63

64

60

60

1.27

58

55

55

51

49

1.52

52

51

48

45

45

1.78

49

49

45

44

42

2.03

51

48

45

38

41

2.29

51

47

45

41

39

2.54

49

47

44

39

38

3.05

49

45

44

41

36

5.08

49

44

42

39

38

Table II

Minimum angle of resolution of a Foucault test object of 94 percent inherent contrast as a function of the exit pupil of a high grade telescopic system.

Exit pupil(millimeters)

Reciprocal of exit pupil

Minimum angle of resolution for two observers (radians ×10^{5})

No. 1

No. 2

0.39

2.6

143

149

0.60

1.7

94

92

1.6

0.62

44

46

3.0

0.33

38

41

4.6

0.22

37

40

Table III

The coefficient of specific resolution for a Foucault test object of 94 percent inherent contrast vs. age of the observer.

Coefficient of specific resolution is defined here as the product of the minimum separable angle (in radians) between the elements of the test object and the aperture (in mm).

Table IV

Computed values of the coefficient of specific resolution of Focault test objects of different inherent contrast viewed through an artificial pupil stop 0.4 of a mm in diameter.

A coefficient of specific resolution
$(\overline{R})$ of 56.0 millimeter-radians ×10^{5} is equivalent to a relative resolving power (RRP) of 100 percent.

Table V

Computed values of the coefficient of specific resolution of Foucault test objects of different inherent contrast viewed through an artificial pupil stop 0.6 of a mm in diameter.

A coefficient of specific resolution
$(\overline{R})$ of 56.0 millimeter-radians ×10^{5} is equivalent to a relative resolving power (RRP) of 100 percent.

Table VI

Computed values of the coefficient of specific resolution of Foucault test objects of different inherent contrast viewed through an artificial pupil stop 0.7 of a mm in diameter.

A coefficient of specific resolution
$(\overline{R})$ of 56.0 millimeter-radians ×10^{5} is equivalent to a relative resolving power (RRP) of 100 percent.

Table VII

Computed values of the coefficient of specific resolution of Foucault test objects of different inherent contrast viewed through an artificial pupil stop 0.8 of a mm in diameter.

A coefficient of specific resolution
$(\overline{R})$ of 56.0 millimeter-radians ×10^{5} is equivalent to a relative resolving power (RRP) of 100 percent.

Table VIII

Computed values of the coefficient of specific resolution of Foucault test objects of different inherent contrast viewed through an artificial pupil stop 1.0 mm in diameter.

A coefficient of specific resolution
$(\overline{R})$ of 56.0 millimeter-radians ×10^{5} is equivalent to a relative resolving power (RRP) of 100 percent.

Table IX

Computed values of the coefficient of specific resolution of Foucault test objects of different inherent contrast viewed through an artificial pupil stop 1.5 mm in diameter.

A coefficient of specific resolution
$(\overline{R})$ of 56.0 millimeter-radians ×10 ^{5} is equivalent to a relative resolving power (RRP) of 100 percent.

Table X

Computed values of the coefficient of specific resolution of Foucault test objects of different inherent contrast viewed through an artificial pupil stop 2.0 mm in diameter.

A coefficient of specific resolution
$(\overline{R})$ of 56.0 millimeter-radians ×10^{5} is equivalent to a relative resolving power (RRP) of 100 percent.

Tables (10)

Table I

Minimum angle of resolution for a Foucault test object of a 94 percent inherent contrast vs. the diameter of an artificial pupil stop placed 3 mm from the pupil of the human eye.

Pupil stop(millimeters)

Minimum angle of resolution (radians ×10^{5}) for five observers

No. 1

No. 2

No. 3

No. 4

No. 5

0.41

144

133

133

137

133

0.46

129

122

125

125

125

0.76

81

76

80

79

77

1.02

67

63

64

60

60

1.27

58

55

55

51

49

1.52

52

51

48

45

45

1.78

49

49

45

44

42

2.03

51

48

45

38

41

2.29

51

47

45

41

39

2.54

49

47

44

39

38

3.05

49

45

44

41

36

5.08

49

44

42

39

38

Table II

Minimum angle of resolution of a Foucault test object of 94 percent inherent contrast as a function of the exit pupil of a high grade telescopic system.

Exit pupil(millimeters)

Reciprocal of exit pupil

Minimum angle of resolution for two observers (radians ×10^{5})

No. 1

No. 2

0.39

2.6

143

149

0.60

1.7

94

92

1.6

0.62

44

46

3.0

0.33

38

41

4.6

0.22

37

40

Table III

The coefficient of specific resolution for a Foucault test object of 94 percent inherent contrast vs. age of the observer.

Coefficient of specific resolution is defined here as the product of the minimum separable angle (in radians) between the elements of the test object and the aperture (in mm).

Table IV

Computed values of the coefficient of specific resolution of Focault test objects of different inherent contrast viewed through an artificial pupil stop 0.4 of a mm in diameter.

A coefficient of specific resolution
$(\overline{R})$ of 56.0 millimeter-radians ×10^{5} is equivalent to a relative resolving power (RRP) of 100 percent.

Table V

Computed values of the coefficient of specific resolution of Foucault test objects of different inherent contrast viewed through an artificial pupil stop 0.6 of a mm in diameter.

A coefficient of specific resolution
$(\overline{R})$ of 56.0 millimeter-radians ×10^{5} is equivalent to a relative resolving power (RRP) of 100 percent.

Table VI

Computed values of the coefficient of specific resolution of Foucault test objects of different inherent contrast viewed through an artificial pupil stop 0.7 of a mm in diameter.

A coefficient of specific resolution
$(\overline{R})$ of 56.0 millimeter-radians ×10^{5} is equivalent to a relative resolving power (RRP) of 100 percent.

Table VII

Computed values of the coefficient of specific resolution of Foucault test objects of different inherent contrast viewed through an artificial pupil stop 0.8 of a mm in diameter.

A coefficient of specific resolution
$(\overline{R})$ of 56.0 millimeter-radians ×10^{5} is equivalent to a relative resolving power (RRP) of 100 percent.

Table VIII

Computed values of the coefficient of specific resolution of Foucault test objects of different inherent contrast viewed through an artificial pupil stop 1.0 mm in diameter.

A coefficient of specific resolution
$(\overline{R})$ of 56.0 millimeter-radians ×10^{5} is equivalent to a relative resolving power (RRP) of 100 percent.

Table IX

Computed values of the coefficient of specific resolution of Foucault test objects of different inherent contrast viewed through an artificial pupil stop 1.5 mm in diameter.

A coefficient of specific resolution
$(\overline{R})$ of 56.0 millimeter-radians ×10 ^{5} is equivalent to a relative resolving power (RRP) of 100 percent.

Table X

Computed values of the coefficient of specific resolution of Foucault test objects of different inherent contrast viewed through an artificial pupil stop 2.0 mm in diameter.

A coefficient of specific resolution
$(\overline{R})$ of 56.0 millimeter-radians ×10^{5} is equivalent to a relative resolving power (RRP) of 100 percent.