Additional Study Notes: Optics Calculation Workbook

Learning objectives

Select one sign convention, convert units before substitution and interpret a calculated image or vergence physically. All examples below were created for this notes package.

Units and optical power

In air, thin-lens power F = 1/f when f is in metres. A +0.25 m focal length corresponds to +4.00 D; a −0.50 m focal length corresponds to −2.00 D. Millimetres must be divided by 1000 and centimetres by 100 before using these relations. A positive converging lens and a negative diverging lens must not be assigned the same sign.

Vergence convention

For this example, light travels left to right, distances to points on the right are positive, and distances to points on the left are negative. Vergence L = n/l, with l in metres. At a thin lens, L′ = L + F.

An object is 0.50 m to the left of a +4.00 D thin lens in air. L = 1/(−0.50) = −2.00 D. Therefore L′ = −2.00 + 4.00 = +2.00 D, and the image is 1/2 = 0.50 m to the right of the lens. The positive emergent vergence indicates converging rays.

Real-is-positive thin-lens convention

A different common convention uses 1/f = 1/do + 1/di, with positive do for a real object and positive di for a real image. It gives the same physical prediction when used consistently. Do not insert a signed Cartesian object distance into this alternative equation without translating conventions.

Example: f = +0.20 m and do = +0.60 m. Then 1/di = 5 − 1.667 = 3.333 m⁻¹, giving di = +0.30 m. Magnification m = −di/do = −0.50. The image is real, inverted and half the object height.

Refraction and plausibility checks

Snell’s law is n1 sin θ1 = n2 sin θ2, with angles measured from the normal. Example: light passes from air, n1 = 1.00, into a medium with n2 = 1.50 at θ1 = 30°. Then sin θ2 = (1/1.50) × 0.5 = 1/3 and θ2 ≈ 19.47°. The ray bends toward the normal, as expected on entry into the higher-index medium.

Before accepting an answer, state its unit, sign and physical meaning. For a sine result exceeding 1, reconsider whether total internal reflection is possible and whether the incident and transmitted indices were assigned correctly.

Study References

• Source notes: Vergence; Refraction; Thin Lenses I and II
• OpenStax: Thin Lenses

External references checked 13 September 2026. Worked numerical examples and teaching activities are original.