Two-lens system image position calculator

It first finds the image made by lens one, converts that location into the signed object distance for lens two, and then calculates the final image. The result includes positions relative to both lenses, total transverse magnification, whether the final image is real or virtual, and whether it is upright or inverted. All distances use meters, and focal lengths may represent converging or diverging lenses.

Describe both lenses with one consistent coordinate system

Enter the signed focal length of each thin lens, the positive distance from the original object to the first lens, and the positive separation between the lenses. A converging lens has a positive focal length, while a diverging lens has a negative focal length. The calculator places lens one at coordinate zero and lens two at the separation coordinate, with light traveling from lens one toward lens two. The original object is therefore a real object on the incoming side of lens one. Keeping every length in meters avoids hidden conversions and makes the returned positions directly comparable. The thin-lens and paraxial assumptions matter: lens thickness is neglected, both lenses share an optical axis, and rays make small angles with that axis. This model is appropriate for classroom ray diagrams, preliminary bench layouts, and first-order optical design. It is not a substitute for thick-lens principal planes, aberration analysis, aperture stops, or wavelength-dependent glass data when those effects materially influence a real instrument.

Follow the intermediate image into the second calculation

The first calculation applies the thin-lens equation to lens one and returns its signed image distance. That image location is then compared with the position of lens two. The signed object distance for lens two equals the lens separation minus the first image distance. If the intermediate image lies to the left of lens two, this distance is positive and lens two receives a real object. If lens one would bring the rays to a focus beyond lens two, the distance is negative: lens two intercepts converging rays and receives a virtual object. This sign change is the central detail that makes a two-lens problem different from two unrelated single-lens calculations. The calculator then applies the same thin-lens equation to lens two. A positive final image distance indicates a real final image on the outgoing side of lens two; a negative value indicates a virtual final image on its incoming side. It also adds the separation to report the final coordinate measured from lens one.

Interpret image position, magnification, and singular cases

Use the final distance from lens two when placing a screen, sensor, or observation plane relative to the second element. Use the final position from lens one when your mechanical drawing has its origin at the first lens. Total transverse magnification is the product of the magnifications produced by the two imaging stages. Its absolute value gives the image-size ratio in this ideal model, while its sign determines orientation: a negative result is inverted and a positive result is upright relative to the original object. The real or virtual label is based on the signed final image distance, not on orientation. Certain exact configurations do not have a finite image position. An object at the focal plane of lens one sends collimated light onward, and an object at the focal plane of lens two produces a final image at infinity. The calculator reports these as invalid finite-position requests rather than returning a non-JSON infinity. It likewise rejects zero focal lengths, nonpositive geometry distances, nonfinite values, and the degenerate case where the intermediate image falls exactly in the second lens plane.