Enter object distance, focal lengths, spacing, and height. See lenswise magnification, image type, and totals. Download tables, graphs, and reports for quick verification tasks.
Use positive focal length for a converging lens and negative focal length for a diverging lens. Distances are in centimeters.
The graph below sweeps object distance to show how total magnification and final image distance change for the current lens settings.
This sample uses fixed values of f1 = 10 cm, f2 = 15 cm, d = 25 cm, and object height = 2 cm.
| u1 (cm) | f1 (cm) | f2 (cm) | d (cm) | v1 (cm) | u2 (cm) | v2 (cm) | Total M | Final height (cm) |
|---|---|---|---|---|---|---|---|---|
| 12 | 10 | 15 | 25 | 60 | -35 | 10.5 | -1.5 | -3 |
| 14 | 10 | 15 | 25 | 35 | -10 | 6 | -1.5 | -3 |
| 18 | 10 | 15 | 25 | 22.5 | 2.5 | -3 | -1.5 | -3 |
| 22 | 10 | 15 | 25 | 18.3333 | 6.6667 | -12 | -1.5 | -3 |
| 30 | 10 | 15 | 25 | 15 | 10 | -30 | -1.5 | -3 |
| 40 | 10 | 15 | 25 | 13.3333 | 11.6667 | -52.5 | -1.5 | -3 |
For lens 1:
1 / f1 = 1 / v1 + 1 / u1
v1 = (f1 × u1) / (u1 - f1)
m1 = -v1 / u1
Object distance for lens 2:
u2 = d - v1
For lens 2:
1 / f2 = 1 / v2 + 1 / u2
v2 = (f2 × u2) / (u2 - f2)
m2 = -v2 / u2
Total magnification:
M = m1 × m2
Final image height:
h′ = h × M
Equivalent focal length of the two-lens system:
1 / Feq = 1 / f1 + 1 / f2 - d / (f1 × f2)
Enter the object distance from the first lens. Use a positive value because the object is placed in front of lens 1.
Enter the focal length of each lens. Use positive values for converging lenses and negative values for diverging lenses.
Enter the separation distance between the two lenses. This value is the center-to-center spacing along the optical axis.
Enter the object height. The calculator uses it to estimate the final image height after both lenses act on the object.
Press Calculate. The page will show the intermediate image from lens 1, the second lens object distance, the final image, and the total magnification.
Use the graph to inspect how changing the first object distance affects the system response. Download the result as CSV or PDF when needed.
A two-lens system forms an image in two stages. Lens 1 creates an intermediate image. That intermediate image then acts as the object for lens 2. Because the second stage depends on the first stage, lens spacing matters greatly.
Magnification in each lens stage equals image distance divided by object distance, with sign included. The sign tells you whether the image is upright or inverted. When you multiply the two stage magnifications, you get the total system magnification.
The intermediate image may be real or virtual. If it forms before the second lens, lens 2 receives a real object. If it forms beyond the second lens, lens 2 receives a virtual object. This changes the second image distance and can flip the image behavior.
Equivalent focal length helps summarize the whole optical pair as one combined system. It is especially useful during early design studies, instrument layout, and magnification planning. Still, the exact image position must be found from the two-step calculation.
This calculator is useful for classroom optics, lab work, telescope sections, relay imaging, and general lens train analysis. It quickly shows how focal length choice, object placement, and lens spacing reshape the final image location and size.
Total magnification is the product of the first and second lens magnifications. It tells you the final image size relative to the original object and whether the image is upright or inverted.
A negative second object distance means the intermediate image lies beyond lens 2. In that case, lens 2 sees a virtual object, which is physically valid in multi-lens optics.
Use a negative focal length for a diverging lens. Use a positive focal length for a converging lens. The sign convention changes image position and magnification results.
Lens 1 would send rays out parallel, so the intermediate image moves to infinity. The calculator blocks that case because the second stage becomes undefined for practical finite output.
A negative magnification means inversion. A positive magnification means the image stays upright relative to the previous stage. The final sign shows the net orientation after both lenses.
No. Equivalent focal length is helpful for quick system behavior, but exact intermediate and final image locations still require the full lens-by-lens calculation used here.
Yes. Lens separation shifts the intermediate image relative to lens 2. Even modest spacing changes can alter the second object distance, final image distance, and total magnification strongly.
Use one consistent length unit for all distance and height inputs. This page is labeled in centimeters, so keeping every input in centimeters gives directly consistent outputs.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.