How this calculation works
Two lenses separated by air do more than add their reciprocal focal lengths. Propagation between them changes both effective power and final image position. A ray-transfer matrix keeps that separation explicit. This page solves the final image for a real object before the first lens and reports the system effective focal length separately. They are different outputs with different reference planes.
System matrix = L2 × T(d) × L1; image distance = −(Au+B)/(Cu+D)A worked example
Two 100 mm converging lenses spaced 50 mm apart have matrix A = 0.5, B = 50 mm, C = −0.015 per mm and D = 0.5. The effective focal length is 66.6667 mm. For an object 300 mm before the first lens, the final image lies 50 mm after the second lens with magnification −0.25. The reference example below uses that complete matrix solution rather than a contact-lens approximation.
Good to know
Only paraxial rays and ideal thin elements are represented. Aperture stops, vignetting, distortion, chromatic behavior and mechanical lens thickness are absent. The output is not a complete optical-system design. A practical instrument also needs verified lens orientation, clear aperture, spacing tolerances and the correct principal-plane information for real elements.
Reference: MIT — paraxial optical systems (opens in a new tab) ↗
What this calculation answers
Two lenses separated by air do more than add their reciprocal focal lengths. Propagation between them changes both effective power and final image position. A ray-transfer matrix keeps that separation explicit.
This page solves the final image for a real object before the first lens and reports the system effective focal length separately. They are different outputs with different reference planes.
The Thin lens real and virtual image magnification calculator handles one ideal lens. Two separated elements require propagation between the planes rather than treating them as one contact power.
Use the right input conventions
Use signed thin-lens focal lengths and the distance between their ideal planes. Both elements share the same surrounding medium. A negative focal length represents a diverging element, while separation is nonnegative.
The object distance is measured before lens one. The final image distance is measured from lens two, positive to its outgoing side. It should not be measured from the first lens or substituted directly for effective focal length.
Interpret the result
The matrix determinant is one in this common-medium paraxial convention. That invariant provides a useful numerical check on the matrix construction but says nothing about optical quality or alignment.
An afocal system can still transform ray directions and image a finite object. Conversely, a particular object position can produce an image at infinity even when the complete system has finite effective focal length.
Work through the example
Two 100 mm converging lenses spaced 50 mm apart have matrix A = 0.5, B = 50 mm, C = −0.015 per mm and D = 0.5. The effective focal length is 66.6667 mm.
For an object 300 mm before the first lens, the final image lies 50 mm after the second lens with magnification −0.25. The reference example below uses that complete matrix solution rather than a contact-lens approximation.
Where the model stops
Only paraxial rays and ideal thin elements are represented. Aperture stops, vignetting, distortion, chromatic behavior and mechanical lens thickness are absent.
The output is not a complete optical-system design. A practical instrument also needs verified lens orientation, clear aperture, spacing tolerances and the correct principal-plane information for real elements.
A few questions, answered
Can I just add the two lens powers?
Only the zero-separation limit uses simple reciprocal focal-length addition in the common medium.
What changes in the comparison?
Bring the lenses into contact and compare final image position and effective power.