A compound lens system consists of two or more lenses used in sequence. Even when each lens is individually simple, the overall system can behave in a more complicated way because the image formed by the first lens becomes the object for the second lens.
This is the basic idea behind many optical instruments, including zoom assemblies, relay optics, and camera lens groups.
For two thin lenses separated by a distance \(d\), a very useful system quantity is the effective focal length. Under the thin-lens approximation, it is given by
\[
\frac{1}{f_{\mathrm{eff}}}
=
\frac{1}{f_1}
+
\frac{1}{f_2}
-
\frac{d}{f_1 f_2}.
\]
Here \(f_1\) and \(f_2\) are the signed focal lengths of the two lenses. A converging lens has positive focal length, while a diverging lens has negative focal length in the sign convention used in this calculator.
The separation term shows immediately that the system is not just “lens 1 plus lens 2.” The spacing matters.
While the effective focal length tells you about the combined optical power of the pair, it does not by itself give the final image position for a particular object.
To find the actual image, the safest method is sequential imaging.
First use the thin-lens equation for lens 1:
\[
\frac{1}{f_1}=\frac{1}{d_{o1}}+\frac{1}{d_{i1}}.
\]
Once \(d_{i1}\) is known, that image becomes the object for lens 2. If the lenses are separated by distance \(d\), then the object distance for lens 2 is
\[
d_{o2}=d-d_{i1}.
\]
This sign matters a lot. If the image from lens 1 lies to the left of lens 2, then \(d_{o2}\) is positive and lens 2 sees a real object.
If the image from lens 1 would lie to the right of lens 2, then \(d_{o2}\) becomes negative and lens 2 sees a virtual object.
That distinction is one of the main reasons compound-lens problems must be handled carefully.
After that, apply the thin-lens equation again to lens 2:
\[
\frac{1}{f_2}=\frac{1}{d_{o2}}+\frac{1}{d_{i2}}.
\]
The value \(d_{i2}\) gives the final image position measured from lens 2. If \(d_{i2}>0\), the final image is real and lies to the right of lens 2. If \(d_{i2}<0\), the final image is virtual and lies to the left of lens 2.
Magnification also occurs in stages. Lens 1 produces magnification
\[
m_1=-\frac{d_{i1}}{d_{o1}},
\]
and lens 2 produces
\[
m_2=-\frac{d_{i2}}{d_{o2}}.
\]
The total magnification is then
\[
m=m_1m_2.
\]
This tells you whether the final image is upright or inverted, and whether it is larger or smaller than the original object.
Another useful quantity is the back focal length, often abbreviated BFL. In a multi-lens system, the rear focal point is not generally one focal length away from the second lens in the naive single-lens sense.
The back focal length measures the distance from the second lens to the final focal point for incoming parallel rays from the left.
This is important in practical lens design because the physical spacing of optics and sensors depends on it.
In the animation for this calculator, two standard rays are traced from the top of the object. One ray begins parallel to the optical axis before lens 1, and the other goes through the center of lens 1.
Both rays then continue to lens 2, where they are refracted again. If the final image is real, the outgoing rays from lens 2 physically meet at the final image point.
If the final image is virtual, the outgoing rays diverge and their backward dashed extensions locate the image.
This sequential construction is the clearest conceptual way to understand compound systems. Even though the effective focal length formula compresses the system into one number,
the image-formation process still happens lens by lens.
That is exactly why two-lens systems are a natural stepping stone toward more advanced topics such as relay optics, zoom systems, principal planes, and matrix methods for three or more optical elements.
