Abstract

The effect of size of pupil of the eye upon the intensity distributions in retinal images of circular apertures observed in Maxwellian view is treated theoretically. Diffraction images of a small source of light are formed in the pupil and are governed by circular apertures whose angular diameters are of the order of minutes of arc. Quantitative data are presented to illustrate the manner in which sharpness of the retinal image of the aperture is increased by increasing the number of diffraction rings that are admitted to the eye. Amplitude and intensity distributions are included for a given diameter of aperture together with various sizes of pupil (including some annular pupils), and for various diameters of aperture together with one size of pupil. Equations are presented for calculating intensity distributions as modified by the Stiles-Crawford effect or by the effect of defocused eyes.

© 1959 Optical Society of America

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References

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  1. M. Born, Optik (Julius Springer, Berlin, 1933).
    [Crossref]
  2. Equation (2) is similar to one developed by H. Osterberg and G. E. Pride [J. Opt. Soc. Am. 40, 14 (1950)] for the case of a microscope objective. The approach of Osterberg and Pride is considerably more general since it starts off without the premise made here, viz., that the geometrical image of S must be in A3. Osterberg and Pride’s elegant development can be shown to lead to Eq. (2) if a number of assumptions are made, including the one that all angles are small enough to be regarded as equal to their sines and tangents. In removing the restriction that S be focused on A3, Osterberg and Pride’s equation retains validity beyond the case of Maxwellian imagery considered in this paper.
    [Crossref]
  3. E. L. O’Neill, IRE Trans. on Inform. Theory. IT2, No. 2, 56 (1956).
    [Crossref]
  4. M. Francon, “Interférences, diffraction et polarisation,” Encyclopedia of Physics, edited by S. Flügge (Springer-Verlag, Berlin, 1956), Vol. XXIV.
  5. This is strictly only valid if P1 is at the first focal plane of lens A2. As a first approximation it can be regarded to hold also for small degrees of simulated ametropia.
  6. H. H. Hopkins, Proc. Roy. Soc. (London) A217, 408 (1953).
  7. S. Shlaer, J. Gen. Physiol. 21, 165 (1937).

1956 (1)

E. L. O’Neill, IRE Trans. on Inform. Theory. IT2, No. 2, 56 (1956).
[Crossref]

1953 (1)

H. H. Hopkins, Proc. Roy. Soc. (London) A217, 408 (1953).

1950 (1)

1937 (1)

S. Shlaer, J. Gen. Physiol. 21, 165 (1937).

Born, M.

M. Born, Optik (Julius Springer, Berlin, 1933).
[Crossref]

Francon, M.

M. Francon, “Interférences, diffraction et polarisation,” Encyclopedia of Physics, edited by S. Flügge (Springer-Verlag, Berlin, 1956), Vol. XXIV.

Hopkins, H. H.

H. H. Hopkins, Proc. Roy. Soc. (London) A217, 408 (1953).

O’Neill, E. L.

E. L. O’Neill, IRE Trans. on Inform. Theory. IT2, No. 2, 56 (1956).
[Crossref]

Osterberg, H.

Pride, G. E.

Shlaer, S.

S. Shlaer, J. Gen. Physiol. 21, 165 (1937).

IRE Trans. on Inform. Theory. (1)

E. L. O’Neill, IRE Trans. on Inform. Theory. IT2, No. 2, 56 (1956).
[Crossref]

J. Gen. Physiol. (1)

S. Shlaer, J. Gen. Physiol. 21, 165 (1937).

J. Opt. Soc. Am. (1)

Proc. Roy. Soc. (London) (1)

H. H. Hopkins, Proc. Roy. Soc. (London) A217, 408 (1953).

Other (3)

M. Born, Optik (Julius Springer, Berlin, 1933).
[Crossref]

M. Francon, “Interférences, diffraction et polarisation,” Encyclopedia of Physics, edited by S. Flügge (Springer-Verlag, Berlin, 1956), Vol. XXIV.

This is strictly only valid if P1 is at the first focal plane of lens A2. As a first approximation it can be regarded to hold also for small degrees of simulated ametropia.

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Figures (4)

Fig. 1
Fig. 1

Simplified version of an optical system employing the principle of Maxwellian view. A point source S is imaged by lenses A1 and A2 in A3, the plane of the pupil of the observer’s eye. The optical system of the latter is represented by the lens in plane A3. P1 is an object plane conjugate to the retina. The retina is represented by plane P2.

Fig. 2
Fig. 2

Left: Amplitude distributions in the plane of the retina when a circular bright disk whose diameter subtends an angle of 2 minutes of arc at the eye is seen in Maxwellian view with pupil diameters of 1, 2, 4, 6, and 8 mm. At the bottom the ideal image is shown for comparison. Ordinates: relative amplitude. Abscissas: nodal point angles in image space expressed in minutes of arc, measured from the center of the image. One minute of arc corresponds to a retinal distance of about 5 μ. Right: relative intensity distributions similarly displayed.

Fig. 3
Fig. 3

Relative retinal intensity distributions for circular apertures of various diameters seen in Maxwellian view with a pupil of 4 mm diameter. Retinal distances are expressed in nodal point angles in minutes of arc, measured from the center of image. “Object diameter” refers to the angular subtense at the eye of the image of the aperture in plane P1.

Fig. 4
Fig. 4

Relative retinal intensity distributions when a circular clear disk 2 minutes of arc in diameter is seen in Maxwellian view with three different annular pupils.

Equations (11)

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u ( r ) = [ K J 1 ( α 0 k r ) ] / α 0 k r
u ( α ) = K k α 0 [ 1 λ 0 2 π exp { - i α k r cos θ } d θ ] J 1 ( α 0 k r ) d r = K α 0 pupil J 0 ( α k r ) J 1 ( α 0 k r ) d r .
u ( α ) = 0 J 0 ( α k r ) J 1 ( α 0 k r ) τ ( r ) d r
τ ( r ) = { 1 if 0 < r < b 0 if r > b .
u ( α ) = 0 b J 0 ( α k r ) J 1 ( α 0 k r ) d r .
u ( α ) = 0 3.26 b J 0 ( α t ) J 1 ( α · t ) d t .
τ ( r ) = { 0 if 0 < r < b 1 1 if b 1 < r < b 2 0 if r > b 2 .
τ ( r ) = { η ( r ) if 0 < r < b 0 if r > b
τ ( r ) = { exp { 1 2 i k Δ F r 2 } if 0 < r < b 0 if r > b .
u ( α ) = 0 3.26 b J 0 ( α t ) J 1 ( α 0 t ) exp { 0.175 i π t 2 Δ F } d t .
u ( α ) · u * ( α ) = [ 0 3.26 b J 0 ( α t ) J 1 ( α 0 t ) × cos ( 0.175 π t 2 Δ F ) d t ] 2 + [ 0 3.26 b J 0 ( α t ) J 1 ( α 0 t ) × sin ( 0.175 π t 2 Δ F ) d t ] 2 .