What is the equality of first and second focal lengths in Sears' Optics?

[ttzhou]

Homework Statement

In Sears' Optics, chapter 4, Sears claims that the first and second focal lengths (distance from first and second focal points to the first and second principal points, respectively) are equal, and he seems to imply that this is true in general, without proof.

I am a mathematician at heart, and the lack of proof of this bothers me immensely. Would any PF'ers be able to shed some light on this matter? It would be greatly appreciated. I did a forum search and went through about 4-5 pages and found nothing similar.

Homework Equations

Basically, $f = f'$ where $f$ stands for the first focal length and $f'$ stands for the second.

The Attempt at a Solution

I tried imagining this using Fermat's principle of reversibility, but it seems kind of sketchy...

 

[Simon Bridge]

What are they calling the "first" and "second" focal lengths?

In general, the front and back focal lengths will be different.
The two are the same if the lens is symmetrical.
See: http://en.wikipedia.org/wiki/Focal_length

 

[Simon Bridge]

What are they calling the "first" and "second" focal lengths?

In general, the front and back focal lengths will be different.
The two are the same if the lens is symmetrical.
See: http://en.wikipedia.org/wiki/Focal_length

 

[ttzhou]

Hi Simon,

As mentioned in my post - the first focal length is defined as the distance from the first (or front) focal point to the first principal point, which is defined as the intersection of the first principle plane and the axis. The second focal length is defined analagously.

I've already read the Wiki article and I did not find it to be of use. My intuition tells me that in general they are not equal, but according to Sears, he constantly makes reference to them being equal, and nowhere has he stated that the lenses are symmetrical. In fact, he does an example in which the lens is asymmetrical, but STILL claims the focal lengths are equal.

 

[Simon Bridge]

OK - I was trying to guide you there , let's try another approach: reading FFL and BFL - under "general optical systems" in wikipedia, there is a detailed derivation showing that these two lengths are not, in general, the same.

Comparing wiki with Sears, however, FFL (eg) is defined from the first optical surface. FL1, by Sears, is defined from the first principle point ... which refines the search: to understand Sears, you need to understand how the principle point is found [pdf]. FL1=|F-P| and FL2=|F'-P'|. Wikipedia calls this "EFL" and, indeed, FL1=FL2.

From there is it a matter of geometry.