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Vergence calculator

Where does a lens put the image? Enter the object distance and lens power, and see every step of V = U + P.

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Vergence calculator

Find where a lens forms an image (V = U + P), or switch between lens power and focal length (P = 1 ÷ f).

Diagram: object on the left of the lens, image shown where the light focuses Light from the object passes through the lens and focuses on the other side. Vergence equals one divided by distance in meters. light → object image
Light travels left to right. The lens is in the middle.
Type a positive number. The calculator applies the minus sign for you (light from an object spreads out).
Plus (+) lenses bring light together; minus (−) lenses spread it out.
Words used here
Diopter (D)
The unit of lens power and of vergence: 1 ÷ distance in meters. 2 D means 1 ÷ 0.5 m.
Vergence
How much light is spreading out or coming together where it meets the lens. Spreading out (diverging) is minus; coming together (converging) is plus; parallel light is 0.
U, P and V
U = vergence of the light arriving from the object. P = lens power. V = vergence of the light leaving the lens.
Sign rule
Light travels left to right. Distances to the left of the lens (in front, where the light comes from) are minus. Distances to the right (behind the lens) are plus.
Real vs. virtual image
A real image forms where light actually comes together (you could catch it on a card). A virtual image is where the light only seems to come from; you see it by looking through the lens.

Works on your device: nothing you type is saved or sent anywhere.

How vergence works

Vergence describes light as it reaches a lens: spreading out (minus), coming together (plus) or parallel (zero). It is measured in diopters, which are 1 ÷ distance in meters.

  1. Object vergence U = 1 ÷ object distance (m). The object is in front of the lens, so the distance is minus: an object 50 cm away gives U = 1 ÷ (−0.50) = −2.00 D.
  2. Add the lens power: V = U + P. With a +5.00 D lens, V = −2.00 + 5.00 = +3.00 D.
  3. Image distance = 1 ÷ V = 1 ÷ 3.00 = 0.333 m. Plus V means a real image 33.3 cm behind the lens; minus V means a virtual image in front of it.

Lens power and focal length are linked the same way: P = 1 ÷ f. A +4.00 D lens focuses parallel light 25 cm behind it; a −2.00 D lens spreads light as if it came from 50 cm in front.

References

  • Schwartz SH. Geometrical and Visual Optics: A Clinical Introduction. McGraw-Hill (chapters on vergence and thin lenses).
  • Keating MP. Geometric, Physical, and Visual Optics. Butterworth-Heinemann (vergence and the thin-lens equation).
  • Brooks CW, Borish IM. System for Ophthalmic Dispensing. Butterworth-Heinemann (lens power and focal length).

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