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Thin lens calculator, conjugation relation and magnification

Image position and nature, magnification, optical power, with the algebraic sign convention set in advance.

Parameters

Measured distance, positive. The algebraic measure OA, negative, is shown with the results.
Positive for a converging lens, negative for a diverging one.
Optional: gives the image height. Same unit as the distances.

Result

Algebraic measure of the image

15

Calculation
1/OA’ − 1/OA = 1/f’: 1/15 − 1/−30 = 1/10
Magnification
γ = OA’ / OA = 15 / −30 = −0.5

The image is real, inverted and reduced. The lens is converging.

Lengths in the input unit.

FF’OOA = −30f’ = 10OA’ = 15
OA = −30

Object and image

Algebraic measure of the object
−30
Lens-to-image distance
15
Magnification
−0.5
Object-to-image distance
45

The lens

Image focal length
10
Optical power
0.1
Notes (3)

Playback only moves the object position: it slides toward the lens, from three focal lengths down to one fifth of one, in eight seconds, and every frame is fully recomputed.

Origin at the optical centre O, positive direction that of the light: a real object measured at 30 cm is written OA = −30.

A screen placed behind the lens catches the image. From a real object, a real image is always inverted.

Calculation, assumptions and sources

OverviewThe Descartes conjugation relation both ways: the image position for a given object, or the lens and its optical power in dioptres from two measured positions.

The Descartes conjugation relation both ways: the image position for a given object, or the lens and its optical power in dioptres from two measured positions. The calculation gives the image position and algebraic measure, the magnification, the optical power, the image height when the object height is known, and the nature of the image (real or virtual, upright or inverted, enlarged or reduced). The entered distance is positive; the algebraic measure, negative for a real object, is shown with the results.

Method & formulas1/OA’ − 1/OA = 1/f’
1/OA’ − 1/OA = 1/f’γ = OA’ / OA = A’B’ / ABV = 1 / f’real object: OA < 0real image: OA’ > 0

The origin is the optical centre, the positive direction that of the light, and positions are algebraic measures. A real object, in front of the lens, has a negative measure.

OA
algebraic measure of the object, negative for a real object.
OA’
algebraic measure of the image. Positive behind the lens, where a screen catches it.
f’
image focal length. Positive for a converging lens, negative for a diverging one.
γ
magnification. Negative, the image is inverted; greater than 1 in absolute value, it is enlarged.
Optical power
1/f’, in dioptres when the focal length is in metres.
Validity domainThe model is the thin lens under Gauss conditions: rays at small angles and close to the optical axis.

The model is the thin lens under Gauss conditions: rays at small angles and close to the optical axis. A thick lens, or one used outside those conditions, shows geometric aberrations that shift and distort the image. The focal length also depends on wavelength (chromatic aberration). An object at the object focus gives an image at infinity: rays leave parallel.