Focal length

All optical imaging systems have a set of two foci, focal lengths, and focal planes. If an object is placed at $F_o$ (i.e., $s_o$ is finite) and is imaged at infinity (i.e., $s_i=\infty$), then this object distance $f_o$ is called the object focal length (or front focal length). From the Gaussian formula (where $n_1$ is the refractive index of the object-side):

$$f_o=\frac{n_{1}R}{n_2-n_1}$$

If an image is formed at $F_i$ (i.e., $s_i$ is finite) when $s_o=\infty$, then this image distance is the image focal length (or back focal length).

$$f_i=\frac{n_{2}R}{n_2-n_i}$$

We collectively refer to $F_o$ and $F_i$ as the foci of the imaging system. In general, $f_i\ne f_o$.

By the sign convention, $f_o$ is positive with a real object (when $F_o$ is to the left of $V$), and $f_i$ is positive with a real image (when $F_i$ is to the right of $V$).

The infinity is not some arbitrary parameter. An object placed at $f_o$ will produce an image at infinity. An object placed infinitely away will image at $f_i$. Consider lens-based systems. This helps us determine the working distances of our lens. For example, camera lenses have a minimum distance the object needs to be away from the lens to be able to focus (front focal length). And there needs a minimum space we need behind the lens/OIS for the image to form.

Parallel rays that form a small angle wrt the OA will converge to an off-axis point (i.e., an off-axis image). The plane perpendicular to the OA where the parallel rays lie is called the image focal plane (or back focal plane). We can similarly define an object focal plane.

The focal length for multiple lenses in contact with each other is given by:

$$\frac{1}{f}=\sum \frac{1}{f_i}$$