Spherical Aberration Calculator

How it is calculated

You enter the ray height h, the distance of an incoming ray from the optical axis, and the mirror's radius of curvature R. The tool applies the small-angle approximation LSA ≈ h² ÷ (4·R) and shows the result in millimetres, together with the working h² ÷ 4R. The paraxial focus of the mirror is at f = R ÷ 2, and the exact shift is (R/2)(1/cosθ − 1) with sinθ = h/R; the shortcut matches it closely for rays near the axis. Treat both inputs as millimetres and use the same unit for both. R must be above zero, h cannot be negative, and h must be smaller than R, since a ray at or beyond the mirror's edge does not hit the sphere; otherwise there is no result. Because the result grows with h², doubling the height makes the aberration four times larger.

Common mistakes

• Entering the focal length where the radius is needed. For a spherical mirror the focal length is about half of R, so typing f in the R box makes the result twice too large.
• Using the mirror's full diameter as h. The formula needs the height of one ray above the axis, so for the edge of a 200 mm mirror use 100.
• Mixing units, such as h in inches and R in millimetres. The result then has no meaning and is still labelled mm.
• Trusting the shortcut for rays far from the axis. With h = 1000 on R = 2000 the tool gives 125 mm, while the exact geometry gives about 154.7 mm; and h equal to or above R shows no result.
• Taking the result as the exact blur size. It is the longitudinal shift of the focus along the axis; the blur seen on the sensor is a transverse quantity and is smaller.