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Free lens maker equation calculator with combined lenses and double lens system ray tracing. Calculate focal length from refractive index & radii of curvature, find equivalent focal length for lens combinations, or trace rays through a two-lens system with step-by-step solutions.
Lens maker equation, combined lenses, or double lens system.
The lens maker's equation relates the focal length of a lens to the refractive index of the material and the radii of curvature of its two surfaces: 1/f = (ฮผ โ 1)(1/Rโ โ 1/Rโ). For a lens immersed in a medium other than air, the generalized form uses ฮผโ/ฮผโ (relative refractive index).
When two thin lenses are placed in contact (d = 0), their powers add directly: P = Pโ + Pโ. Equivalent focal length: 1/F = 1/fโ + 1/fโ. Used in compound eyepieces and corrective lenses.
When lenses are separated by distance d: 1/F = 1/fโ + 1/fโ โ d/(fโยทfโ). The separation term changes the effective power. Telescopes and microscopes rely on precise lens spacing.
A converging crown glass lens paired with a diverging flint glass lens. Net effect is converging but chromatic aberration is minimized. Essential in camera lenses and telescopes.
Image from lens 1 becomes the object for lens 2. Requires solving the thin lens equation twice. Total magnification = mโ ร mโ. Used in microscopes, projectors, and relay optics.
Rโ > 0, Rโ < 0. Always converging (f > 0). Most common lens shape.
Rโ < 0, Rโ > 0. Always diverging (f < 0). Used in myopia correction.
Rโ = โ, Rโ < 0. Converging. Common in laser optics and condensers.
Rโ = โ, Rโ > 0. Diverging. Used in beam expanders.
Both surfaces curve same way, Rโ > 0, Rโ > 0 (Rโ < Rโ). Net converging. Used in eyeglasses.
Both surfaces curve same way, Rโ < 0, Rโ < 0. Net diverging. Less common.