Page 334 - Modern Optical Engineering The Design of Optical Systems

Optical System Layout 313

marginal ray at the image cannot exceed 90°. This limits the numeri-
cal aperture of the system to NA n′ sin 90° n′; for systems in air
with distant sources the limiting relative aperture becomes f/0.5.
There are other limits imposed on the speed of the objective lens; the
design of the system may be incapable of whatever resolution is
required at large aperture ratios, or physical limitations (or predeter-
mined relationships) may limit the acceptable speed of the objective.
We can introduce the effective f/# of the objective by multiplying
both sides of Eq. 13.19 by A; setting (f/#) F/A and rearranging, to
get, for systems in air,
D
(f/#) (13.20)
2A
or for systems with the final image in a medium of index n′
A
NA n′ sin u′ (13.21)
D
Equation 13.21 can also be demonstrated by setting the optical
invariant (Eq. 4.14) at the objective (I A /2) equal to the invariant at
the image (I
Dn′u′) and substituting sin u′ for u′ (in accordance
1
2
with our requirement for aplanatism).
Since the (f/#) cannot be less than 0.5 and sin u′ cannot exceed 1.0,
it is apparent that the objective aperture A, half-field angle , and
detector size D, are related by
A
! 1.0 (13.22)
n′D

It should be noted that Eq. 13.22, since it can be derived by way of
the optical invariant with no assumptions as to the system between
object and detector, is valid for all types of optical systems, including
reflecting and refracting objectives with or without field lenses,
immersion lenses, light pipes, etc. It is thus quite futile to attempt a
design with the left member of Eq. 13.22 larger than unity; in fact, it is
sometimes difficult to exceed (efficiently) a value of 0.5 when good
imagery is required. This limit is applicable to any optical system, no
matter how simple or complex. Equation 13.22 is exactly analogous to
Eq. 12.24 for projection or illumination systems.
As an example of the application of Eq. 13.22, let us determine the
largest field of view possible for a radiometer with a 5-in aperture and
a 1-mm (0.04-in) detector. If the detector is in air (n′ 1.0) we then
have, from Eq. 13.22,

5
! 1.0 or ! 0.008 radians
0.04