**[»] Optics**.

Today’s post will be brief and is a short though on the concept of paraxial image position.

Almost all students learn someday that the object and image position of a paraxial thin lens are bound together by the formula

where *i* is the image position, *o* the object position and *f* the focal length of the lens.

Some have also heard about Newton’s formulation of the same law

with *x=o-f* and *x’=i-f*.

From either of these, you may compute the position of the image plane knowing the object plane or *vice-versa*.

You may wonder where do these formulas come from and how general they are or how they have to be modified to become more general.

Let us take the [»] generic system *M* such that

and we will add two distances corresponding to the image and object plane positions

We find

for which we know that if we trace any ray that originates on axis *(0,u)* we should find an exit ray that is also on axis *(0,u’)* because of the imaging condition:

The condition is met if

for which we are not interested in the trivial solution *u=0* and therefore we look for

After rearrangement we obtain

which is precisely Newton’s formula:

I will keep however the two following relations which are more convenient to find the value or *i* and *o* in the context of our #DevOptical series:

Now, restricting ourselves to the use of thin paraxial lenses (that is *EFFL=FFL=BFL*), we find

that we can rearrange into

and

which is the paraxial focus formula that we all know!

In conclusion, Newton’s formula is the most general expression, and the paraxial focus formula should only be used for thin-lens systems. They all refer to paraxial conditions obviously.

That is all for today!

I would like to give a big thanks to **James**, **Lilith**, **Cam**, **Samuel**, **Themulticaster**, **Sivaraman**, **Vaclav** and **Arif** who have supported this post through [∞] Patreon. I also take the occasion to invite you to donate through Patreon, even as little as $1. I cannot stress it more, you can really help me to post more content and make more experiments!

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[»] #DevOptical Part 2: Paraxial Raytracing and the ABCD Matrix

[»] #DevOptical Part 19: A Quick Focus Algorithm

[»] #DevOptical Part 16: Partition Ratios Revisited