Page 76 - Modern Optical Engineering The Design of Optical Systems

Optical System Considerations 59

Multiplying by yy p and rearranging, we get
y nu ynu y n′u′ yn′u′
p p p p
Note that on the left side of the equation the angles and indices are for
the left side of the surface (that is, before refraction) and that on the
right side of the equation the terms refer to the same quantities after
refraction. Thus y p nu ynu p is a constant which is invariant across
any surface.
By a similar series of operations based on Eq. 3.17, we can show that
(y p nu ynu p ) for a given surface is equal to (y p nu ynu p ) for the next
surface. Thus this term is not only invariant across the surface but also
across the space between the surfaces; it is therefore invariant through-
out the entire optical system or any continuous part of the system.

Invariant Inv y nu ynu n (y u yu ) (4.14)
p p p p

The invariant and magnification
As an example of its application, we now write the invariant for the
object plane and image plane of Fig. 4.4. In an object plane y p h,
n n, y 0, and we get

Inv hnu (0) nu hnu
p
In the corresponding image plane y p h′, n n′, y 0, and we get
Inv h′n′u′ (0) n′u′ h′n′u′
p
Equating the two expressions gives

hnu h′n′u′ (4.15)
which can be rearranged to give a very generalized expression for the
magnification of an optical system
h′ nu
m (4.16)
h n′u′
Equation 4.16 is, of course, valid only for the extended paraxial
region; this relationship is sometimes applied to trigonometric calcu-
lations, where it takes the form of Eq. 4.17 for the magnification at a
zone of the aperture.
hn sin u h′n′ sin u′ (4.17)
n sin u
or m 5
nrsin ur