Two-Lens Optical System Calculator

Explore image formation in a two-lens system such as a simple slide projector. Both lenses may be converging or diverging, so you can investigate the four sign combinations: ++, −+, +− and −−. The calculator keeps the signs of image and object distances so real and virtual images are handled correctly.

Current lens combination: L1 + / L2 + f > 0: converging lens f < 0: diverging lens

Optical layout

The diagram updates after calculation. The horizontal axis is the optical axis; distances are shown schematically rather than to exact scale.

object lens image Dashed image marker = virtual image

Stage 1

L1 forms the intermediate image

Focal length of lens L1 [mm]Use a positive value for a converging lens, negative for a diverging lens.

Object distance from L1, dₒ₁ [mm]For the initial slide/object this distance is positive.

Sign convention: f₁ > 0 → converging; f₁ < 0 → diverging. A positive dᵢ₁ means a real intermediate image to the right of L1; a negative value means a virtual image to the left of L1.

Intermediate-image results

Intermediate image distance dᵢ₁

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Magnification m₁

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Enter values and calculate.

Stage 2

L2 forms the final image

Focal length of lens L2 [mm]Use a positive value for a converging lens, negative for a diverging lens.

Distance between L1 and L2, L [mm]The position of the intermediate image determines the signed object distance for L2.

For L2: dₒ₂ = L − dᵢ₁. The intermediate image from L1 is the object for L2. A positive dₒ₂ means that this object is on the input side of L2 (a real object for L2); a negative dₒ₂ means that it is on the output side (a virtual object for L2).

Final-image results

Object distance for L2, dₒ₂

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Final image distance dᵢ₂

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Magnification m₂

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Total magnification M

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Calculate Stage 1 first.

System summary

The total magnification is meaningful when both stages have finite image/object distances. Special cases such as an intermediate image at infinity are described explicitly.

Lens combination

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Intermediate image type

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Final image type

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Final image position from L1

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Calculation method and sign convention

Thin-lens equation: 1/f = 1/dₒ + 1/dᵢ

Image distance: dᵢ = f dₒ / (dₒ − f)

Linear magnification: m = −dᵢ / dₒ

Second-lens object distance: dₒ₂ = L − dᵢ₁

Total magnification: M = m₁ × m₂, when both stages have finite distances

Positive dᵢ means a real image on the outgoing-light side of that lens. Negative dᵢ means a virtual image. Positive magnification means upright; negative means inverted. For L2, dₒ₂ is signed: positive when the object point is on the input side of L2 (real object for L2), and negative when it is on the output side (virtual object for L2). The object for L2 is the intermediate image formed by L1, which may itself be real or virtual. A lens with f > 0 is converging; a lens with f < 0 is diverging.

Special case: when dₒ₁ = f₁ for a converging L1, the emerging light is collimated and the intermediate image is at infinity. L2 can still form a finite image at dᵢ₂ = f₂, but the overall linear magnification relative to the original slide is not defined by m₁ × m₂.

Model limitation: thin-lens and paraxial approximation; lens thickness, aberrations, diffraction and finite-aperture effects are not included.

Two-Lens Optical System Calculator

Optical layout

System summary

Two-lens slide projector and classroom optics calculator

What can be explored

Lambda Physics optical calculators