Transcript Cardinal planes/points in paraxial optics

Stops in optical systems
(5.3)
Hecht 5.3
Monday September 30, 2002
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Stops in Optical Systems
In any optical system, one is concerned with a number of things
including:
1. The brightness of the image
Image of S
formed at
the same
place by
both lenses
S
Bundle of
rays from S,
imaged at S’
is larger for
larger lens
Two lenses of the same
focal length (f), but
diameter (D) differs
S’
More light collected
from S by larger
lens
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Stops in Optical Systems


Brightness of the image is determined primarily
by the size of the bundle of rays collected by the
system (from each object point)
Stops can be used to reduce aberrations
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Stops in Optical Systems
How much of the object we see is determined by:
(b) The field of View
Q
Q’
(not seen)
Rays from Q do not pass through system
We can only see object points closer to the axis of the system
Field of view is limited by the system
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Theory of Stops

We wish to develop an understanding of
how and where the bundle of rays are
limited by a given optical system
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Aperture Stop
A stop is an opening (despite its name) in
a series of lenses, mirrors, diaphragms,
etc.
 The stop itself is the boundary of the lens
or diaphragm
 Aperture stop: that element of the optical
system that limits the cone of light from
any particular object point on the axis of
the system

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Aperture Stop (AS)
O
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Entrance Pupil (EnP)
is defined to be the image of the aperture stop in all the lenses
preceding it (i.e. to the left of AS - if light travels left to right)
E’
How big does the
aperture stop look
E
to someone at O
L1
E’E’ = EnP
O
F1’
EnP – defines the
cone of rays
accepted by the
system
E
E’
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Exit Pupil (ExP)
The exit pupil is the image of the aperture stop in the lenses
coming after it (i.e. to the right of the AS)
E’’
E
L1
E”E” = ExP
F2’
O
E
E’’
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Location of Aperture Stop (AS)
In a complex system, the AS can be found
by considering each element in the system
 The element which gives the entrance
pupil subtending the smallest angle at the
object point O is the AS

Example, Telescope
Objective=AS=EnP
eyepiece
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Example: Eyepiece
f1’ = 6 cm
f2’ = 2 cm
E
O
E
9 cm
1 cm
Ф1 = 1 cm
3 cm
ФD = 1 cm
Ф2 = 2 cm
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Example: Eyepiece
Find aperture stop for s = 9 cm in front of L1.
To do so, treat each element in turn –
find EnP for each
(a) Lens 1 – no elements to the left
tan µ1 = 1/9
defines cone of rays accepted
O
µ1
1 cm
9 cm
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Example: Eyepiece
Find aperture stop for s = 9 cm in front of Diaphragm. Find EnP
(b) Diaphragm – lens 1 to the left
Look at the system
E E’
from behind the slide
1 1 1
 
1 s' 6
s'  1.2cm
O
M 
E
9 cm
s'
 1.2
s
E’
1 cm
1.2 cm
ФD’ = 1.2 cm
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Example: Eyepiece
Calculate maximum angle of cone of rays accepted by entrance
pupil of diaphragm
E’
O
µ2
0.6 cm
9 + 1.2 cm
tan µ2 = 0.6/10.2 ≈ 1/17
E’
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Example: Eyepiece
(c) Lens 2 – 4 cm to the left of lens 1
Look at the system
from behind the slide
1 1 1
 
4 s' 6
s'  12cm
O
M 
s ' 12

3
s 4
Ф2’ = 6 cm
9 cm
4 cm
Ф2 = 2 cm
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Example: Eyepiece
Calculate maximum angle of cone of rays accepted by entrance
pupil of lens 2.
3 cm
O
µ3
9 + 12 cm
tan µ3 = 3/21 = 1/7
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Example: Eyepiece
Component
Entrance pupil
Acceptance cone
angle
(degrees)
Lens 1
tan µ1 = 1/9
6.3
Diaphragm
tan µ2 = 1/17
3.4
Lens 2
tan µ3 = 1/7
8.1
Thus µ2 is the smallest angle
The diaphragm is the element that limits the cone of rays
from O
Diaphragm = Aperture Stop
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Example: Eyepiece
Entrance pupil is the image of the diaphragm in L1.
E’
µ2 = tan-1 (1/17)
O
E’
EnP
9 cm
1.2 cm
ФD’ = 1.2 cm
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Example: Eyepiece
Exit pupil is the image of the aperture stop (diaphragm) in L2.
f2’ = 2 cm
E
1 1 1
 
3 s' 2
s'  6cm
O
M 
E
9 cm
1 cm
s'
6
   2
s
3
Ф2’ = 2 cm
3 cm
ФD = 1 cm
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Example: Eyepiece
E’
f2’ = 2 cm
E
ФExP’ = 2 cm
O
ExP
E
3 cm
ФD = 1 cm
6 cm
E’
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Chief Ray
for each bundle of rays, the light ray which
passes through the centre of the aperture
stop is the chief ray
 after refraction, the chief ray must also
pass through the centre of the exit and
entrance pupils since they are conjugate
to the aperture stop
 EnP and ExP are also conjugate planes of
the complete system

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Marginal Ray

Those rays (for a given object point) that
pass through the edge of the entrance and
exit pupils (and aperture stop).
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Chief Ray from T
•Proceed toward centre of
EnP
Ray tracing with pupils and stops
•Refracted at L1 to pass
though centre of AS
•Refracted at L2 to pass
(exit) through centre of ExP
T
L2
L1
Q’’
P’
P
O’
O
Marginal Rays from T,O
•Must proceed towards edges of
EnP
•Refracted at L1 to pass through
edge of AS
•Refracted at L2 to pass (exit)
through ExP.
T’
Q
Q’
P’’
AS EnP
ExP
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Exit Pupil
Defines the bundle of rays at the image
Q’’
T
O’
α’
O
T’
P’’
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Field Stop

That component of the optical system that
limits the field of view
A
θ
d
A = field of view at distance d
θ = angular field of
view
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