Page 201 - Modern Optical Engineering The Design of Optical Systems

184 Chapter Nine

Another way of expressing this relationship is by the numerical
aperture (usually abbreviated as N.A. or NA), which is the index of
refraction (of the medium in which the image lies) times the sine of the
half angle of the cone of illumination.
Numerical aperture NA n′ sin U′ (9.2)

Numerical aperture and f-number are obviously two methods of
defining the same characteristic of a system. Numerical aperture is
more conveniently used for systems that work at finite conjugates
(such as microscope objectives), and the f-number is appropriately
applied to systems for use with distant objects (such as camera lenses
and telescope objectives). For aplanatic systems (i.e., systems corrected
for coma and spherical aberration) with infinite object distances, the
two quantities are related by:

1
f-number (9.3)
2NA
The terms “fast” and “slow” are often applied to the f-number of an
optical system to describe its “speed.” A lens with a large aperture
(and thus a small f-number) is said to be fast, or to have a high speed.
A smaller aperture lens is described as slow. This terminology derives
from photographic usage, where a larger aperture allows a shorter (or
faster) exposure time to get the same quantity of energy to the film and
may allow a rapidly moving object to be photographed without blurring.
It should be apparent that a system working at finite conjugates will
have an object-side numerical aperture as well as an image-side
numerical aperture and that the ratio NA/NA′ (object-side NA)/
(image-side NA) must equal the absolute value of the magnification.
The term “working f-number” is sometimes used to describe the
numerical aperture in f-number terms. If we use the terms “infinity
f-number” for the f-number defined in Eq. 9.1, then the image-side
working f-number is equal to the infinity f-number times (1 m),
where m is the magnification.
Another term that is occasionally encountered is the T-stop, or
T-number. This is analogous to the f-number, except that it takes into
account the transmission of the lens. Since an uncoated, many element
lens made of exotic glass may transmit only a fraction of the light that
a low-reflection coated lens of simpler construction will transmit, such
a speed rating is of considerable value to the photographer. The rela-
tionship between f-number, T-number, and transmission is
f-number
T-number (9.4)

transmission