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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 = (μ − 1)(1/R₁ − 1/R₂). For a lens immersed in a medium other than air, the generalized form uses μ₁/μ₂ (relative refractive index).
When two thin lenses are placed in contact (d = 0), their powers add directly: P = P₁ + P₂. Equivalent focal length: 1/F = 1/f₁ + 1/f₂. Used in compound eyepieces and corrective lenses.
When lenses are separated by distance d: 1/F = 1/f₁ + 1/f₂ − d/(f₁·f₂). 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 = m₁ × m₂. Used in microscopes, projectors, and relay optics.
R₁ > 0, R₂ < 0. Always converging (f > 0). Most common lens shape.
R₁ < 0, R₂ > 0. Always diverging (f < 0). Used in myopia correction.
R₁ = ∞, R₂ < 0. Converging. Common in laser optics and condensers.
R₁ = ∞, R₂ > 0. Diverging. Used in beam expanders.
Both surfaces curve same way, R₁ > 0, R₂ > 0 (R₁ < R₂). Net converging. Used in eyeglasses.
Both surfaces curve same way, R₁ < 0, R₂ < 0. Net diverging. Less common.