a shutter speed of 1/100 second and an f/stop of f/5.6, the exposure time is cut in half while the irradiance (E0/32) is doubled, thereby leaving no net change in the film exposure (J/m2).

The numerical aperture is another important design parameter for a lens, related directly to how much light the lens gathers. If the focal length of a design lens increases and its diameter decreases, the solid angle (cone) of useful light rays from object to image for such a lens decreases. For example, the concept of a numerical aperture finds immediate application in the design of the objective lens (the lens next to the specimen under observation) for a microscope, as we show below. Light-gathering capability is crucial for microscopes.

Figure 3-24 depicts the light-gathering power of a lens relative to a point O on a specimen covered by a glass slide. Lens L is the objective lens of a microscope focused on the specimen. On the right side of the symmetry axis of the lens, the light-gathering power of the lens with air between the cover slide and the lens is depicted in terms of half-angle ɑair. On the left side, by contrast, the increased light-gathering power of the lens with oil situated between the cover slide and the lens is shown in terms of the larger half-angle ɑoil. The oil is chosen so as to


have an index of refraction (n0) very near that of the cover slide (ng) so that little or no refraction occurs for limiting ray 2 at the glass-oil interface. Consequently the half-angle oil is greater than the half-angle air. As Figure 3-24 shows, ray 1 suffers refraction at the glass-air interface, thereby restricting the cone of rays accepted by the lens to the smaller half-angle air.

The numerical aperture of a lens is defined so as to exhibit the difference in solid angles (cones) of light accepted, for example, by an “oil-immersion” arrangement versus an air-immersion setup.

The definition of numerical aperture (N.A.) is given in Equation 3-9 as
where n is the index of refraction of the intervening medium between object and lens and ɑ is the half-angle defined by the limiting ray (ɑair or ɑoil in Figure 3-24). The “light-gathering” power of the microscope’s objective lens is thus increased by increasing the refractive index of the intervening medium.