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FUNDAMENTAL OPTICS
Optical Coatings
& Materials
IMAGING PROPERTIES OF LENS SYSTEMS
THE OPTICAL INVARIANT
To understand the importance of the NA, consider its
relation to magnification. Referring to figure 4.6,
sin v
=
NA
.
When a lens or optical system is used to create an image
of a source, it is natural to assume that, by increasing the
diameter (Ø) of the lens, thereby increasing its CA, we
will be able to collect more light and thereby produce a
brighter image. However, because of the relationship
EXAMPLE: SYSTEM WITH FIXED INPUT NA
Two very common applications of simple optics involve
coupling light into an optical fiber or into the entrance
slit of a monochromator. Although these problems
appear to be quite different, they both have the same
limitation – they have a fixed NA. For monochromators,
this limit is usually expressed in terms of the f-number. In
addition to the fixed NA, they both have a fixed entrance
pupil (image) size.
Suppose it is necessary, using a singlet lens, to couple
the output of an incandescent bulb with a filament 1 mm
in diameter into an optical fiber as shown in figure 4.7.
Assume that the fiber has a core diameter of 100 mm and
an NA of 0.25, and that the design requires that the total
distance from the source to the fiber be 110 mm. Which
lenses are appropriate?
By definition, the magnification must be 0.1. Letting s+s”
total 110 mm (using the thin-lens approximation), we can
use equation 4.3,
f =m
(s + s ″)
,
(m + )
Machine Vision Guide
The magnification of the system is therefore equal to
the ratio of the NAs on the object and image sides of
the system. This powerful and useful result is completely
independent of the specifics of the optical system, and
it can often be used to determine the optimum lens
diameter in situations involving aperture constraints.
To understand how to use this relationship between
magnification and NA, consider the following example.
Gaussian Beam Optics
NA
m =NA .
NA. ″
m =
NA ″ (4.15)
SAMPLE CALCULATION
Fundamental Optics
=
Since the NA of a ray is given by CA/2s, once a focal
length and magnification have been selected, the value
of NA sets the value of CA. Thus, if one is dealing with
a system in which the NA is constrained on either the
object or image side, increasing the lens diameter
beyond this value will increase system size and cost but
will not improve performance (i.e., throughput or image
brightness). This concept is sometimes referred to as the
optical invariant.
Optical Specifications
s″
NA
sarrive
v v ″ NA
″ s s″ sin
NA ″
sin
we m
at
Since
simply
== is
=. th .
s sin
NA
s
s ″v ″″ NA ″
simply
magnification
of
the
system,
Since
ofthe
thesystem,
system,
Since NA
isissimply
tehthe
emagnification
magnification
of
hhe
m sarrive
=″ s at .
we
NA
″
Sincewe arrive
is
simply
t
we arrive
s at at
Material Properties
CA
NA (object side) = sin v = CA (4.10)
NA (object side) = sin v = s
sCA
NA″ (image side) = sin v ″ = CA
NA″ (image side) = sin v ″ = s ″ (4.11)
s ″
w NA (object side) = sin v = CA
s
w
which can be rearranged to show
CA
NAcan
(object
side) = sinto
v =show
which
be rearranged
CA
which
bevside)
arranged
CA
can
s sin
NA″=(image
= sin vto″ show
=s
s ″
CA = s sin v
andNA″ (image vside) = sin v ″ = CA
w
CA
andNA (object side)
v ″ = sin v = s ″
CA can
=tosbe
″ sin
leading
which
rearranged
to show
s
(4.12)
=
s
sin
v
CA
w CA
″ = sin v =
NA =(object
leading
tos ″ sinside)
sCA
sNA″
v rearranged
NA = sintov ″show
sinbe
″ can
CA
and
which
=CA
==(image
= vside)
.
s vsin
sand
NA
sin
″
s ″
side)″ =
sin v = CA
s =NA
v ″ = NA
sin(object
vNA
CA
=sin
sv″″sin
″ ″=. sin v ″ =s
NA″
(image
side)
s
w
CA
s″
s″
Since
simply
the magnification
of the system,
leading
toissbe
NA″
(image
side)
= sin
v show
″=
and
s
″
(4.13)
v
=
sin
″
″
which
can
rearranged
to
s is simply the magnification
w
s ″ of the system,
Since
s sin v rearranged
sw″ can
CA
we
arrive
which
leading
CA
s sin=v NA . to show
==toatbe
swhichsin
v ″be rearranged
NA ″
we arrive
at
to show
NA
can
s vsin
vv NA
s ″ = sin
andCA
leading
to
m =
.
=
.
sCA
″ NA
=
s
sin
v
NA
s ==sin
″″ NA
Since
is vsimply
th″
m
and
CA
leading
tos ″ sin. ″
andss″ NA ″
v he magnification of the system,
CA
= sin
iss ″simply
″ th
Since
sleading
vsin v NA
″ CA
=s =tos ″=sin ″ (4.14)
.
we arrive
atv ″ NAhe
leading
s to
sin
″ magnification of the system,
between magnification and NA, there can be a theoretical limit beyond which increasing the diameter has no
effect on light-collection efficiency or image brightness.
(see eq. 4.3)
to determine that the focal length is 9.1 mm. To
determine the conjugate distances, s and s”, we utilize
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Imaging Properties of Lens Systems
A99
FUNDAMENTAL OPTICS
and
Fundamental Optics
equation 4.6,
s ( m + ) = s + s ″, (see eq. 4.6)
and find that s = 100 mm and s” = 10 mm.
We can now use the relationship NA = CA/2s or
NA”= CA/2s” to derive CA, the optimum clear aperture
(effective diameter) of the lens.
With an image NA of 0.25 and an image distance (s”) of
10 mm,
. =
CA = mm.
Accomplishing this imaging task with a single lens
therefore requires an optic with a 9.1 mm focal length
and a 5 mm diameter. Using a larger diameter lens will
not result in any greater system throughput because
of the limited input NA of the optical fiber. The singlet
lenses in this catalog that meet these criteria are LPX5.0-5.2-C, which is plano-convex, and LDX-6.0-7.7-C and
LDX-5.0-9.9-C, which are biconvex.
Making some simple calculations has reduced our choice
of lenses to just three. The following chapter, Gaussian
Beam Optics, discusses how to make a final choice of
lenses based on various performance criteria.
CA
s″
s
CA
2
v″
v
CA
image side
object side
Figure 4.6 Numerical aperture and magnification
magnification = h″ = 0.1 = 0.1!
h
1.0
filament
h = 1 mm
NA =
optical system
f = 9.1 mm
CA
= 0.025
2s
NA″ =
CA
= 0.25
2s ″
CA = 5 mm
fiber core
h″ = 0.1 mm
s = 100 mm
s″ = 10 mm
s + s″ = 110 mm
Figure 4.7 Optical system geometry for focusing the output of an incandescent bulb into an optical fiber
A100
Imaging Properties of Lens Systems
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