Thin lens and mirror equation for image distance, object distance or focal length, with magnification; Snell's law refraction and critical angle.
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With f = 10 cm and the object 30 cm away, the image forms 15 cm beyond the lens: real, inverted, reduced (m = −0.5).
Real is positive: distances to real objects and images are positive, virtual ones negative; converging lenses and concave mirrors have f > 0. For a lens a real image forms on the far side.
The thin-lens equation, 1/f = 1/dₒ + 1/dᵢ, links the focal length f, the object distance dₒ and the image distance dᵢ, and the same equation holds for spherical mirrors. Give any two and the calculator solves the third, then finds the magnification m = −dᵢ/dₒ, the image height, whether the image is real or virtual, and the power in dioptres (1/f with f in metres). Snell's law, n₁ sin θ₁ = n₂ sin θ₂, gives the refracted angle at a boundary and the critical angle beyond which light is totally reflected.
With the defaults, a converging lens of f = 10 cm and an object 30 cm away, the image forms 15 cm behind the lens, inverted and half size (m = −0.5), the arrangement in a camera. Light passing from air into water at 45° bends to 32.05°.
Signs follow the real-is-positive convention: virtual images have negative dᵢ, and diverging lenses and convex mirrors have negative f. Lenses are thin, so thickness and aberrations are ignored.
Snell's law, n₁ sin θ₁ = n₂ sin θ₂, gives a ray's angle after it crosses into a new medium, with both angles measured from the normal. From air (n ≈ 1) into glass (n = 1.5) at 30°, sin θ₂ = 0.5 ÷ 1.5 = 0.333, so the ray continues at 19.47°.
The thin lens equation, 1/f = 1/dₒ + 1/dᵢ, locates the image formed by a lens or curved mirror. A 10 cm converging lens with an object 30 cm away forms a real image 15 cm behind it, inverted and half the object's size.
The normal is the line perpendicular to the surface. The lens equation assumes a thin lens and rays close to its axis.
The refractive index n = c/v compares the speed of light in a vacuum, c, with its speed in the material, v. Glass with n = 1.5 slows light to about 2.0 × 10⁸ m/s, two-thirds of its vacuum speed.
"Index of refraction" and "refraction coefficient" usually name this same n. Fresnel reflection and transmission coefficients are different quantities: they describe how much of the light is reflected or passes through, not which way it bends.
Typical indices for yellow light at 589 nm:
The index changes with wavelength, which is why a prism spreads white light into colors. Use the value for the wavelength you care about.
Light entering a higher index bends toward the normal, and light entering a lower index bends away from it. A ray at 0° passes straight through, and equal indices give no bending at any angle (OpenStax, refraction).
Angles are rounded to six decimal places, and the pi/6 rad row is 30° written in radians. "None" means there is no critical angle, because the light enters an equal or higher index.
| Example | n₁ | n₂ | Incidence | Refraction | Critical angle |
|---|---|---|---|---|---|
| index concept | 1 | 1.5 | 30 deg | 19.471221° | None |
| total reflection | 1.5 | 1 | 45 deg | TIR | 41.810315° |
| default water | 1.000293 | 1.333 | 45 deg | 32.047229° | None |
| normal incidence | 1 | 1.5 | 0 deg | 0° | None |
| equal indices | 1.5 | 1.5 | 45 deg | 45° | None |
| grazing equal indices | 1.5 | 1.5 | 90 deg | 90° | None |
| radian incidence | 1 | 1.5 | pi/6 rad | 19.471221° | None |
| below critical | 1.5 | 1 | 30 deg | 48.590378° | 41.810315° |
| exact critical | 2 | 1 | 30 deg | 90° | 30° |
| just above critical | 2 | 1 | 30.0001 deg | TIR | 30° |
Going from a higher index to a lower one, the critical angle is θc = asin(n₂/n₁). From glass (1.5) into air (≈ 1), θc = asin(0.6667) = 41.81°. Past that angle Snell's law would need sin θ₂ > 1, so no light crosses and all of it reflects back.
At exactly the critical angle the refracted ray runs along the surface, at 90°. In the last two rows, 30° from a medium of index 2 just grazes out, and 30.0001° is reflected.
Into air, the critical angle is about 48.6° for water and 24.4° for diamond. Diamond's small critical angle traps light inside a cut stone, and the same effect keeps light inside an optical fiber (OpenStax, total internal reflection).
For f = 10 cm and dₒ = 30 cm, 1/dᵢ = 1/10 − 1/30 = 2/30, so dᵢ = 15 cm. The magnification is m = −dᵢ/dₒ = −15/30 = −0.5: a 2 cm object gives a 1 cm image, upside down.
Lens power is P = 1/f with f in meters. A 10 cm lens has P = 1/0.1 m = +10 diopters, the unit used on eyeglass prescriptions.
Move the object inside the focal length, to 5 cm, and 1/dᵢ = 1/10 − 1/5 = −1/10. The image distance is −10 cm: a virtual, upright image twice the size (m = +2). That is how a magnifying glass works.
At dₒ = 2f (20 cm) the image also sits at 20 cm, inverted and the same size (m = −1). With the object exactly at the focal point, the rays leave parallel and no finite image forms.
These rules follow the real-is-positive convention:
Enter the focal length as a size without a sign; the chosen lens or mirror type supplies it. A virtual image distance is entered as a negative number.
When you know both distances and solve for f, the sign identifies the element. An object at 30 cm with a virtual image at −5 cm gives 1/f = 1/30 − 1/5, so f = −6 cm, which only a diverging lens or convex mirror can produce.
Distances are in centimeters except in the "metre result" row, which uses meters. Magnification has no unit, power is in diopters, and values are rounded to six decimal places.
| Example | f | dₒ | dᵢ | m | Power D |
|---|---|---|---|---|---|
| default lens | 10 | 30 | 15 | -0.5 | 10 |
| inside focal length | 10 | 5 | -10 | 2 | 10 |
| diverging lens | -10 | 30 | -7.5 | 0.25 | -10 |
| concave mirror | 10 | 30 | 15 | -0.5 | 10 |
| convex mirror | -10 | 30 | -7.5 | 0.25 | -10 |
| solve object | 10 | 30 | 15 | -0.5 | 10 |
| solve focal | 10 | 30 | 15 | -0.5 | 10 |
| opposite focal sign warning | -6 | 30 | -5 | 0.166667 | -16.666667 |
| metre result | 0.1 | 0.3 | 0.15 | -0.5 | 10 |
A concave mirror and a converging lens with the same f put the image at the same distance. A diverging lens or convex mirror always forms a virtual, upright, reduced image of a real object, as the −7.5 cm rows show (OpenStax, thin lenses).
Refraction keeps the frequency and shortens the wavelength to λ/n: 589 nm light is about 442 nm inside water. To convert between wavelength, frequency and photon energy, use the wavelength calculator.
Checked against: 1/dᵢ = 1/10 − 1/30 ⇒ dᵢ = 15 cm, m = −½; P = 1/0.1 m
Checked against: 1/dᵢ = 1/10 − 1/5 = −1/10 ⇒ dᵢ = −10 cm
Checked against: 1/dᵢ = −1/10 − 1/20 = −3/20 ⇒ dᵢ = −20/3 cm
Checked against: 1/dᵢ = 1/10 − 1/20 ⇒ dᵢ = 20 cm, m = −1 (OpenStax UP3 §2.3)
1/f = 1/dₒ + 1/dᵢ, where f is the focal length, dₒ the object distance and dᵢ the image distance, all measured from the lens. For f = 10 cm and an object at 30 cm, 1/dᵢ = 1/10 − 1/30, so dᵢ = 15 cm. A negative dᵢ means a virtual image on the same side as the object, as seen through a magnifying glass.
m = −dᵢ/dₒ, and the image height is m times the object height. A negative m means the image is inverted; |m| above 1 means it is enlarged. An object 5 cm from a 10 cm magnifying glass gives dᵢ = −10 cm and m = +2, an upright virtual image twice the size. At dₒ = 2f the image is real, inverted and the same size (m = −1).
θc = arcsin(n₂/n₁), which exists only when light heads into a medium with a lower index (n₁ > n₂). Into air it is 41.8° from glass (n = 1.5), 48.6° from water (1.333) and 24.4° from diamond (2.417), which is why cut diamonds sparkle. A ray meeting the surface at more than θc from the normal is reflected completely, the principle behind optical fibre.
n₁ sin θ₁ = n₂ sin θ₂: the refractive index times the sine of the angle from the normal is the same on both sides of a boundary. Light entering water (n = 1.333) from air at 45° continues at 32.0°, bent towards the normal because water has the higher index. Going the other way the ray bends away from the normal, and beyond the critical angle it cannot leave at all.
Power P = 1/f with f in metres, measured in dioptres (D). A converging lens with f = 10 cm has P = +10 D, and a diverging lens with f = −50 cm has −2 D. Spectacle prescriptions use this unit: short sight is corrected with negative (diverging) lenses and long sight with positive ones, and the powers of thin lenses in contact add.
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