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

![](/images/articles/-devoptical-part-12--the-paraxial-image-position-formula/image001.png)

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

![](/images/articles/-devoptical-part-12--the-paraxial-image-position-formula/image002.png)

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](/article/-devoptical-part-2--paraxial-raytracing-and-the-abcd-matrix/) *M* such that

![](/images/articles/-devoptical-part-12--the-paraxial-image-position-formula/image003.png)

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

![](/images/articles/-devoptical-part-12--the-paraxial-image-position-formula/image004.png)

 

We find

![](/images/articles/-devoptical-part-12--the-paraxial-image-position-formula/image005.png)

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:

![](/images/articles/-devoptical-part-12--the-paraxial-image-position-formula/image006.png)

The condition is met if

![](/images/articles/-devoptical-part-12--the-paraxial-image-position-formula/image007.png)

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

![](/images/articles/-devoptical-part-12--the-paraxial-image-position-formula/image008.png)

After rearrangement we obtain

![](/images/articles/-devoptical-part-12--the-paraxial-image-position-formula/image009.png)

![](/images/articles/-devoptical-part-12--the-paraxial-image-position-formula/image010.png)

![](/images/articles/-devoptical-part-12--the-paraxial-image-position-formula/image011.png)

![](/images/articles/-devoptical-part-12--the-paraxial-image-position-formula/image012.png)

which is precisely Newton’s formula:

![](/images/articles/-devoptical-part-12--the-paraxial-image-position-formula/image013.png)

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:

![](/images/articles/-devoptical-part-12--the-paraxial-image-position-formula/image014.png)

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

![](/images/articles/-devoptical-part-12--the-paraxial-image-position-formula/image015.png)

that we can rearrange into

![](/images/articles/-devoptical-part-12--the-paraxial-image-position-formula/image016.png)

and

![](/images/articles/-devoptical-part-12--the-paraxial-image-position-formula/image017.png)

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](https://www.patreon.com/thepulsar). 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!