Simple Lens Systems

1
Simple Lens Systems

• Two Thin Lenses in Air

2
Thin Lens in Air
3
Two Thin Lenses in Air
Adding a second positive lens results in greater
overall convergence
4
Two Thin Lenses in Air

• One way to track the light path through the
  system of two lenses is in terms of incident and
  refracted vergences at lens 1 (L1 , L?1) and lens
  2 (L2 , L?2).
• The two lenses will focus parallel light in a
  location (F?e ) determined by combined lens power
  and separation (d)

5
Two Thin Lenses in Air
6
The Equivalent Lens

• The system of two thin lenses can be replaced by
  an equivalent (single) thin lens that will focus
  parallel incident light at the same location (F
  ?e ).
• The equivalent lens will also produce the same
  (refracted) vergence in the plane of lens 2 (L?2).

7
Two Thin Lenses in Air
8
Equivalent Lens - Definitions

• Equivalent Power (Fe ) power of the single thin
  lens that replaces the system of two lenses
• Second Equivalent Focal Length (f??e ) second
  focal length of the equivalent lens
• Second Principal Plane ( H ) position of the
  equivalent lens relative to image space

9
Vertex and Equivalent Powers

• Back Vertex Power (F?V ) equal to refracted
  vergence of light emerging from the second lens
  of the two lens system (L?2)
• Back Vertex Focal Length (f??V ) distance from
  the second lens (back vertex) in the system to
  the second equivalent focus (F?e )

(in air)
10
Vertex and Equivalent Powers
Equivalent Lens
Back vertex focal length is measured between two
physically definable locations the back vertex
(second lens) of the system and the second
equivalent focus, F?e (located with a movable
screen).
11
Vertex and Equivalent Powers
Equivalent Lens
Second equivalent focal length is measured to the
physically definable F?e , but its origin at the
second principal plane (H?) cannot be seen or
physically located.
12
Vertex and Equivalent Powers
Equivalent Lens
Back vertex power is therefore of greater
practical value. Spectacle lenses are specified
by back vertex power. Lensometers measure F?V of
spectacle lenses
13
Calculating Back Vertex Power
Back Vertex Power (F?V ) equals L?2 (refracted
vergence emerging from lens 2) for parallel
incident light at F1
14
Calculating Back Vertex Power

• Back Vertex Focal Length (f??V ) distance from
  the second lens (back vertex) in the system to
  the second equivalent focus (F?e )

15
Calculating Back Vertex Power
Back Vertex Focal Length (f??V ) distance from
the second lens (back vertex) in the system to
the second equivalent focus (F?e )
16
Calculating Back Vertex Power

• F?V is the refracted vergence of light at F2 for
  parallel incident light at F1

• Using step vergences at lens 1

17
Two Thin Lenses in Air
18
Vertex Power

• Next define incident vergence at lens 2 in terms
  of the path of the refracted ray emerging from
  lens 1

19
Two Thin Lenses in Air
20
Vertex Power
Now refract through F2
21
Two Thin Lenses in Air
22
Deriving an Expression for Back Vertex Power
23
Back Vertex Power
24
Back Vertex Power
25
Back Vertex Power
26
Ray Path through Thin Lens System
27
Ray Path through Equivalent Lens f?e vs f?V
28
Equivalent Power
29
Similar Triangles to F?1 (2nd Focus of Lens 1)
30
Similar Triangles to F?1
31
Equivalent Power
32
Equivalent Power
33
Front Vertex Power (Fv) and First Principal Plane
(H)

• Sending parallel light backward through the lens
  system locates the first equivalent focus (Fe )
• Front vertex focal length (FV) is measured from
  lens 1 to Fe.
• The first principal plane (H) is the position of
  the equivalent lens that will replace the lens
  system with respect to rays from (or traveling
  to) object space

34
Thick Lens Theory Principal Planes
35
Positions of Principal Planes (H and H?)
36
Finding Principal Plane Positions
e distance from lens 1 to H
e? distance from lens 2 to H?
37
Finding First Principal Plane Position from Lens
Powers
38
Finding Second Principal Plane Position from Lens
Powers
39
Example 1.13 Two Thin Lenses in Air
Find principal plane positions for the following
system
40
Example 1.13 Two Thin Lenses in Air
Vertex Powers
41
Positions of Principal Planes (H and H?)
42
Example 1.13 Two Thin Lenses in Air
Second Principal Plane (H?)
43
Positions of Principal Planes (H and H?)
44
Ray-Tracing Lens Systems (in air)
45
Properties of Principal Planes

• Principal planes are conjugate planes of unit
  linear magnification
• Placing an object (theoretically) at H will
  result in an identical image (same size and
  orientation) at H?
• In practical terms, an object striking H at
  height H1 will simultaneously leave H? at the
  same height H?1
• Optically, the space between H and H? does not
  exist

46
Ray-Tracing Principal Planes
47
Properties of Principal Planes

• Rays are incident from object space at H
• They are then translated to H? at exactly the
  same height
• All refraction appears to occur at H?
• This is because H? is the part of the equivalent
  lens that relates to image space

48
Ray-Tracing Equivalent Lenses The Abstract

• An equivalent lens is no more difficult to
  ray-trace than a thin lens. This is because it is
  a thin lens.
• The thin lens just happens to be in two different
  locations simultaneously at H to receive
  incident rays, then at H? to refract rays into
  image space
• Despite the physical separation of H and H? there
  is no optical separation
• The difficult part about ray-tracing equivalent
  lenses is locating H and H?

49
Example 1.14 Vergence Relation and
Magnification with Equivalent Lenses
Using the lens system from example 1.13, find
image position and magnification for a real
object 40 cm in front of the front vertex
When dealing with an equivalent lens, all
distances in object space are measured from H
50
Example 1.14 Vergence Relation and
Magnification with Equivalent Lenses
The image is therefore located (9.95 cm e?)
from lens 2
Linear Magnification
The image is inverted and diminished in size. It
is located 9.45 cm from the back vertex of the
system