Focal Length Calculator

The focal length is derived from the thin lens equation by solving for f:

$$\frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i} \implies f = \frac{d_o \cdot d_i}{d_o + d_i}$$

This formula accepts signed values for d_i: positive for real images and negative for virtual images. The resulting focal length sign indicates the optical element type:

• f > 0: Converging element (convex lens or concave mirror)
• f < 0: Diverging element (concave lens or convex mirror)

The radius of curvature for a mirror is related to the focal length by:

$$R = 2f$$

This arises because the focal point of a spherical mirror lies halfway between the mirror surface and the center of curvature. For lenses, the relationship between focal length and radii of curvature involves the refractive index (see the Lens Maker Equation Calculator).

The optical power in diopters is:

$$P = \frac{1}{f_{\text{meters}}}$$

Diopters are the standard unit used in optometry for prescribing corrective lenses. A +2.0 D lens has f = 50 cm and is converging (for farsightedness), while a −3.0 D lens has f = −33.3 cm and is diverging (for nearsightedness).

In experimental optics, determining focal length from conjugate distances (d_o and d_i pairs) is a standard technique. Multiple measurements can be averaged for accuracy, and plotting 1/d_i versus 1/d_o yields a straight line whose intercepts give 1/f.

The focal length is the distance from the optical center to the focal point. Positive values indicate converging optics, negative values indicate diverging optics. The radius of curvature R = 2f applies directly to spherical mirrors. For lenses, it provides an equivalent mirror radius.

The optical power in diopters (D) is the reciprocal of focal length in meters. Higher absolute values mean stronger converging or diverging action. Optometrists use diopters because powers of multiple thin lenses in contact simply add: P_total = P₁ + P₂ + P₃.