1. Transverse Magnification
Basic Optics, Chapter 20

2. Transverse Magnification
Let’s talk about transverse magnification
Also known as lateral or linear magnification
Transverse mag concerns the relative height of objects and images in our ray tracings

3. Transverse Magnification
Let’s talk about transverse magnification
Also known as lateral or linear magnification
Transverse mag concerns the relative height of objects and images in our ray tracings
In principle, with careful tracing, one could simply measure the image and object and determine the ratio directly
Fortunately, there are less tedious methods

4. Transverse Magnification
Transverse magnification is defined as:
Image height
Object height F1 F2
Thin plus lens N
Object height
Image height

5. Transverse Magnification
Transverse magnification is defined as:
Image height
Object height
OK, but how do we determine object
and image heights when all we have
(usually) is info re vergence? F1 F2
Thin plus lens N
Object height
= ?
Image height
= ?

6. Transverse Magnification
Thin plus lens F1 N F2
Here is a ray tracing from a previous chapter.

7. Transverse Magnification
Thin plus lens F1 N F2
Here is a ray tracing from a previous chapter.
Here it is with only the nodal ray and lens axis ray drawn.

8. Transverse Magnification
Thin plus lens F1 N F2
Here is a ray tracing from a previous chapter.
Here it is with only the nodal ray and lens axis ray drawn.
Think back to high-school geometry—what does the
figure look like?

9. Transverse Magnification
Thin plus lens q F1 N F2 q
Similar triangles

10. Transverse Magnification
Thin plus lens O q F1 N F2 q I u v

11. Transverse Magnification
Thin plus lens O q F1 N F2 q I u v
Transverse magnification = I/O (by definition)

12. Transverse Magnification
Thin plus lens O q F1 N F2 q I u v
Transverse magnification = I/O (by definition)
By similar triangles: I/O = v/u

13. Transverse Magnification
Thin plus lens
Therefore, transverse magnification is
determinable by simply taking a ratio
of the image distance to the object distance O q F1 N F2 q I u v
Transverse magnification = I/O (by definition)
By similar triangles: I/O = v/u

14. Transverse Magnification
Thin plus lens
Therefore, transverse magnification is
determinable by simply taking a ratio
of the image distance to the object distance O q F1 N F2 q I u v
Transverse magnification = I/O (by definition)
By similar triangles: I/O = v/u
But we can make it more convenient still…

15. Transverse Magnification
The Vergence Formula
Recall the
Vergence
Formula…
Vergence
contributed
by the lens
Vergence of
incoming light
Vergence of
light leaving lens
U + P = V u v
u = 1/U
v = 1/V

16. Transverse Magnification
The Vergence Formula
Recall the
Vergence
Formula…
Vergence
contributed
by the lens
Vergence of
incoming light
Vergence of
light leaving lens
U + P = V
…and the relationship
between vergence
(big U, big V)
and distance
(little u, little v) u v
u = 1/U
v = 1/V

17. Transverse Magnification
Thin plus lens
U+P=V O q F1 N F2 q I u v
SO, transverse magnification = I/O (by definition)
AND, by similar triangles, I/O = v/u
AND, by the Vergence Formula, v/u = =
1/V
1/U U V
u = 1/U
v = 1/V

18. Transverse Magnification
Thin plus lens
U+P=V O q F1 N F2 q I u v
SO, transverse magnification = I/O (by definition)
AND, by similar triangles, I/O = v/u
AND, by the Vergence Formula, v/u = =
I/O = U/V
THEREFORE,
1/V
1/U U V
u = 1/U
v = 1/V

19. So, in summary: Transverse Magnification U+P=V u v F1 N F2
Thin plus lens
U+P=V
Object height F1 N F2
Image height u v

20. Transverse Magnification
Image height
Object height
Transverse magnification is defined as:
Thin plus lens
U+P=V
Object height F1 N F2
Image height u v

21. Transverse Magnification
Image height
Object height
Transverse magnification is defined as:
Transverse magnification is equal to:
Thin plus lens
U+P=V
Object height F1 N F2
Image height u v

22. Transverse Magnification
Image height
Object height
Transverse magnification is defined as:
Transverse magnification is equal to:
(By similar triangles)
Image distance (v)
Object distance (u)
Thin plus lens
U+P=V
Object height F1 N F2
Image height u v

23. Transverse Magnification
Image height
Object height
Transverse magnification is defined as:
Transverse magnification is equal to:
(By the Vergence Law)
(By similar triangles)
Vergence of incoming light (U)
Vergence of light leaving lens (V)
Image distance (v)
Object distance (u)
Thin plus lens
U+P=V
Object height F1 N F2
Image height u v

24. Transverse Magnification
Image height
Object height
Transverse magnification is defined as:
Transverse magnification is equal to:
(By the Vergence Law)
(By similar triangles)
Vergence of incoming light (U)
Vergence of light leaving lens (V)
Image distance (v)
Object distance (u)
A few final points about transverse magnification: O I F1 N F2 F2 N F1 I O

25. Transverse Magnification
Image height
Object height
Transverse magnification is defined as:
Transverse magnification is equal to:
(By the Vergence Law)
(By similar triangles)
Vergence of incoming light (U)
Vergence of light leaving lens (V)
Image distance (v)
Object distance (u)
A few final points about transverse magnification:
--The sign of the value indicates the relative orientations of object and image O I F1 N F2 F2 N F1 I O

26. Transverse Magnification
Image height
Object height
Transverse magnification is defined as:
Transverse magnification is equal to:
(By the Vergence Law)
(By similar triangles)
Vergence of incoming light (U)
Vergence of light leaving lens (V)
Image distance (v)
Object distance (u)
A few final points about transverse magnification:
--The sign of the value indicates the relative orientations of object and image
--A positive value indicates the image has the same orientation as the object
(i.e., both are either above or below the lens axis) O I F1 N F2 F2 N F1 I O

27. Transverse Magnification
Image height
Object height
Transverse magnification is defined as:
Transverse magnification is equal to:
(By the Vergence Law)
(By similar triangles)
Vergence of incoming light (U)
Vergence of light leaving lens (V)
Image distance (v)
Object distance (u)
A few final points about transverse magnification:
--The sign of the value indicates the relative orientations of object and image
--A positive value indicates the image has the same orientation as the object
(i.e., both are either above or below the lens axis)
--A negative value indicates they are on opposite sides of the lens axis O O F1 N F2 F2 N F1 I I

28. Transverse Magnification
Image height
Object height
Transverse magnification is defined as:
Transverse magnification is equal to:
(By the Vergence Law)
(By similar triangles)
Vergence of incoming light (U)
Vergence of light leaving lens (V)
Image distance (v)
Object distance (u)
A few final points about transverse magnification:
--The size of the value indicates the relative size of object and image

29. Transverse Magnification
Image height
Object height
Transverse magnification is defined as:
Transverse magnification is equal to:
(By the Vergence Law)
(By similar triangles)
Vergence of incoming light (U)
Vergence of light leaving lens (V)
Image distance (v)
Object distance (u)
A few final points about transverse magnification:
--The size of the value indicates the relative size of object and image
--Transverse mag > 1 Image is larger than the object I O
Transverse mag = 2 F1 N F2

30. Transverse Magnification
Image height
Object height
Transverse magnification is defined as:
Transverse magnification is equal to:
(By the Vergence Law)
(By similar triangles)
Vergence of incoming light (U)
Vergence of light leaving lens (V)
Image distance (v)
Object distance (u)
A few final points about transverse magnification:
--The size of the value indicates the relative size of object and image
--Transverse mag > 1 Image is larger than the object
--Transverse mag < 1 Image is smaller than the object O
Transverse mag = .5 I F1 N F2

31. Transverse Magnification
Image height
Object height
Transverse magnification is defined as:
Transverse magnification is equal to:
(By the Vergence Law)
(By similar triangles)
Vergence of incoming light (U)
Vergence of light leaving lens (V)
Image distance (v)
Object distance (u)
A few final points about transverse magnification:
--The size of the value indicates the relative size of object and image
--Transverse mag > 1 Image is larger than the object
--Transverse mag < 1 Image is smaller than the object
--Transverse mag = 1 Image and object are the same size O
Transverse mag = -1 F1 N F2 I