by Michael Tidwell
The lens and cornea of the eye focus the collimated ray bundles to a spot on the retina. The size of the spot formed on the retina is inversely proportional to the resolution of the system. A smaller focused spot allows the possibility of more resolvable spots spread over the same unit of area on the retina. The number of resolvable lines in the display, N, is given by:
where Df = the angular extent of each spot and q = the instantaneous field of view. For any display, a higher number of resolvable points corresponds to higher information content in the display.
Diffraction Limit
The diffraction limit of light is an optical phenomena which, for a circular aperture, sets a lower bound on the diameter of the focused circular spot. The minimum spot diameter D, of a beam passing through an optical system is,
where l = wavelength of the light, f = the focal length of the optical system, and a = the radius of the of the system's limiting aperture. This minimum spot diameter is called the Airy disk and represents the diameter of the first dark fringe in the concentric fringe pattern, a result of interference, of monochromatic light passing through a circular aperture. For the eye, with a pupil diameter of 2 [mm] in bright green light [3], the Airy disk diameter is
The angular extent of a spot on the retina is
where f_{eye} = the focal length of the eye. For the above bright green light adaptation example, the angular extent of the focused spot is
Optical Aberrations
A perfect point spot can only be formed by convergent spherical rays. Imperfect lenses and mirrors and even the eye introduce wavefront aberrations to the light waves in any optical system. The most common types of aberration in an achromatic optical system are [2,24]:
1) Spherical aberration - caused by a change in focal length as a function of lens radius.
2) Coma - caused by the difference in lens power between the far (top) and near (bottom) reaches of a lens for an off axis point (centered below optic axis). Coma causes a comet shaped point to be formed instead of a circular one.
3) Astigmatism - caused by an optical system with a given focal length in one plane, say parallel to the earth (sagittal plane), and a different focal length in the orthogonal plane (tangential plane).
4) Field curvature - caused when the focal plane of the system is curved and not flat.
Chromatic aberrations occur in polychromatic systems and arise from lens performance which is dependent on the wavelength of light passing through the system. The two most common chromatic aberrations are [3]:
1) Longitudinal chromatic aberration - Occurs when the focal length of a lens is wavelength dependent. The result of longitudinal chromatic aberration is a different focal plane location longitudinally for each wavelength passing through the system. The difference in focal length for different colors is due to the refractive index of the material being wavelength dependent.
2) Lateral color - caused by the off-axis imaging characteristics of the lens being
wavelength dependent. The result of lateral color is an off axis polychromatic point being imaged to a rainbow of different image height locations for different wavelengths.
where I = retinal illuminance, "pupil area" refers to the area of the pupil of the eye, and
R = the effectivity ratio. The effectivity ratio, R, allows for the Stiles-Crawford effect and is,
where d = the eye's pupil diameter in millimeters. As shown by dimensional analysis on the equation for I , trolands reduce effectively to the units of optical power per unit steradian.
The Stiles-Crawford effect describes the contribution to brightness sensation of light entering different points of the pupil (i.e. light entering the center of the pupil contributes more to the sensation of brightness than does light entering farther from the pupil center). Some standard scene luminance values, L, and their corresponding Stiles-Crawford corrected estimated retinal illuminance values, I, are given in Table II.1 [36,37].
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Measurements of display image quality depend heavily on two display characteristics, resolution and "contrast" (see subsequent sections). It is virtually fruitless to discuss image quality in terms of either resolution or "contrast" without including the other. Definitions for display resolution, contrast, contrast ratio, and modulation contrast are given in Sections 2.4.1-2.4.4. Whenever possible, the meanings of the terms are related to the effect or result at the retina.
where f_{eye} is the focal length of the eye. Display resolution is often measured in cycles per degree for periodic gratings such as bar patterns or sinusoidal gratings.
where L_{Dmax} = the maximum display luminance and L_{Dmin} = the minimum display luminance. Extending the definition of contrast in terms of estimated retinal illuminance gives
where I_{Dmax} = the maximum estimated retinal illuminance due to the display and I_{Dmin} = the minimum estimated retinal illuminance due to the display. In other words, the values of I_{Dmax} and I_{Dmin} correspond to the estimated retinal illuminance values of displays with luminance values of L_{Dmax} and L_{Dmin} respectively. In the case of a retinal scanning display, as in this thesis, estimated retinal illuminance is a preferable measure of display brightness as there is no screen in the system.
where L_{Dmax} = the maximum display luminance and L_{Dmin} = the minimum display luminance. Extending the definition of contrast in terms of estimated retinal illuminance gives
where I_{Dmax} = the maximum estimated retinal illuminance due to the display and I_{Dmin} = the minimum estimated retinal illuminance due to the display. The values of I_{Dmax} and I_{Dmin} correspond to the estimated retinal illuminance values for displays with luminance values of L_{Dmax} and L_{Dmin} respectively.
where L_{Dmax} = the maximum display luminance and L_{Dmin} = the minimum display luminance. Extending the definition of contrast in terms of estimated retinal illuminance gives
where I_{Dmax} = the maximum estimated retinal illuminance due to the display and I_{Dmin} = the minimum estimated retinal illuminance due to the display. In other words, the values of I_{Dmax} and I_{Dmin} correspond to the estimated retinal illuminance values of displays with luminance values of L_{Dmax} and L_{Dmin} respectively.