Lens power in diopters is 1 ÷ focal length in meters. Image vergence equals object vergence plus lens power: V = U + P, and the image sits 1 ÷ V meters from the lens. A real object gives a negative U. Example: an object 50 cm before a +5.00 D lens images 33.3 cm behind it.
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The formula lists on ABO exam formulas and NCLE exam formulas give the equations. This guide explains the signs and works through the cases that trip people up.
Diopters and focal length
P = 1 ÷ f, with f in meters (or P = 100 ÷ f in cm). A +4.00 D lens focuses parallel light 25 cm behind it. A −4.00 D lens spreads it as if it came from a point 25 cm in front.
Sign convention
Light travels left to right. Distances measured against the light (to the left of the lens) are negative; distances with the light (to the right) are positive. A real object in front of the lens sends diverging light, so U is negative. This matches our vergence calculator, where you enter the object distance and it uses U = −100 ÷ distance in cm.
V = U + P
Image vergence V is the object vergence U plus the lens power P. Then the image distance is 1 ÷ V meters. Positive V means a real image behind the lens; negative V means a virtual image in front; V = 0 means the light leaves parallel and the image is at infinity.
Hand-solve each example, then check it in the vergence calculator.
Worked examples
- Converging lens. Object 50 cm in front of +5.00 D: U = −100 ÷ 50 = −2.00. V = −2.00 + 5.00 = +3.00. Image 100 ÷ 3 = 33.3 cm behind the lens, real.
- Diverging lens. Object 25 cm in front of −2.00 D: U = −4.00. V = −6.00. Image 100 ÷ −6 = −16.7 cm, virtual.
- Object at infinity. +4.00 D: U = 0. V = +4.00. Image 25 cm behind the lens, the focal point.
- Object at the focal point. 20 cm in front of +5.00 D: U = −5.00. V = 0. The light leaves parallel, so the image is at infinity.
- Object inside the focal length. 10 cm in front of +5.00 D: U = −10.00. V = −5.00. Image 20 cm in front, virtual and magnified: how a magnifier works.
Where vergence leads
Move a lens and the vergence it delivers to the eye changes; that is the vertex distance formula. Reading at 40 cm needs 100 ÷ 40 = 2.50 D of focusing, which the eye supplies by accommodating; the accommodation calculator uses the same idea.
Practice problems
Original problems. Use the sign convention above.
A lens has a focal length of +40 cm. What is its power?
Show answer
100 ÷ 40 = +2.50 D
A −8.00 D lens: what is its focal length?
Show answer
100 ÷ −8.00 = −12.5 cm (virtual focal point in front of the lens)
Object 40 cm in front of a +10.00 D lens. Where is the image?
Show answer
U = −100 ÷ 40 = −2.50. V = −2.50 + 10.00 = +7.50. Image = 100 ÷ 7.50 = 13.3 cm behind the lens (real)
Object 1 m in front of a +3.00 D lens. Where is the image?
Show answer
U = −1.00. V = +2.00. Image = 50 cm behind the lens
Object 50 cm in front of a −1.00 D lens. Where is the image?
Show answer
U = −2.00. V = −3.00. Image = 100 ÷ −3.00 = −33.3 cm, virtual, on the object side
Frequently asked questions
What is vergence in optics?
Vergence describes how light is spreading out or coming together at a point, in diopters. It equals 1 divided by the distance in meters to where the light came from or is heading. Diverging light is negative and converging light is positive.
How do you convert focal length to diopters?
Divide 1 by the focal length in meters, or 100 by the focal length in centimeters. A 40 cm focal length is 100 ÷ 40 = +2.50 D; a −12.5 cm focal length is −8.00 D.
What does a negative image vergence mean?
The light leaving the lens is still diverging, so it never comes to a real focus. The image is virtual and sits on the same side as the object, at 1 ÷ V meters.
Is vergence on the ABO exam?
ABO-NCLE’s test specifications list Ophthalmic Optics as 25% of the ABO exam. They don’t list individual formulas, so we can’t say how often vergence itself appears; it is the basis for lens power, vertex and accommodation math.
Source: ABO-NCLE — ABO (NOCE) test specifications (opens in a new tab)
Official sources
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