Radius of Curvature (R)

Lens Diameter (d)

Lens Sag (s)

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Aperture Radius (r)

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How the SAG Calculator Works

The sagitta (SAG) of a spherical optical surface is the axial depth from the vertex of the curved surface down to the plane of its clear aperture. It is the fundamental parameter that connects the radius of curvature and the aperture diameter of a lens — without it, center thickness, edge thickness and minimum blank size cannot be determined.

Diagram of a spherical lens surface showing the sagitta (SAG), radius of curvature R, half-diameter r and the axial depth from vertex to aperture plane — **Figure 1:** Cross-section of a spherical surface showing the sagitta s, radius of curvature R and half-diameter r. The SAG is the axial distance from the surface vertex to the chord defined by the clear aperture edge.

\[ s = R - \sqrt{R^2 - r^2} \]

Exact sagitta formula

Where:

s Sagitta (SAG) : the axial depth of the surface in mm.
R Radius of curvature : the radius of the sphere that best fits the surface in mm.
r Half-diameter : the semi-aperture, equal to half the clear aperture diameter D.

This is the exact formula, valid for any ratio of aperture to radius. The paraxial approximation s ≈ D² / 8R is commonly used in first-order design but introduces significant error when the aperture is large relative to the radius — typically when D/2R exceeds 0.1. This calculator always uses the exact formula.

Sign convention: For a convex surface the SAG is positive. For a concave surface the SAG is negative. The magnitude is identical for both, only the direction of the depth changes.

Why Use a SAG Calculator in Optical Design?

The sagitta (SAG) of a lens surface is the axial distance from the vertex of a curved surface to the plane of its clear aperture. It is one of the most fundamental parameters in optical fabrication — without knowing the SAG, it is impossible to determine the correct center thickness, edge thickness, or blank size needed to grind and polish a lens to its specified radius of curvature.

The exact SAG formula is derived from the geometry of a sphere. For a surface with radius of curvature R and semi-diameter y (half the clear aperture D), the sagitta is:

SAG = R − √(R² − y²)

The paraxial approximation SAG ≈ D² / 8R is commonly used in first-order optical design, but introduces significant error for fast surfaces (low f-number) where D/2R exceeds 0.1. This SAG calculator uses the exact formula.

Key takeaways

SAG determines lens blank thickness — the minimum center or edge thickness required before grinding begins.
Larger radius of curvature = smaller SAG = flatter surface. Smaller R = deeper curve = larger SAG.
SAG scales with D² — doubling the aperture quadruples the sagitta for the same radius of curvature.
Convex and concave surfaces have the same SAG magnitude for the same R and D — only the sign convention differs.
Aspheric surfaces use an extended SAG equation with a conic constant K and higher-order polynomial terms.

Where SAG Calculations Matter

1. Lens blank sizing

Before grinding a spherical surface, the optician must ensure the blank is thick enough to accommodate the SAG on both surfaces. For a biconvex lens, the minimum center thickness equals the sum of both surface SAGs plus a machining allowance. Underestimating SAG leads to scrapped blanks.

2. Center and edge thickness

SAG directly links center thickness to edge thickness via the lensmaker geometry. For a positive lens, center thickness = edge thickness + SAG(R1) + SAG(R2). Optical drawings tolerance one dimension and reference the other, both are meaningless without accurate SAG values.

3. Interferometric testing

When testing a spherical surface with a Fizeau interferometer, the test sphere or transmission flat must be positioned at the center of curvature. The SAG tells the optician how far to translate the reference surface along the optical axis to null the fringes. Getting this wrong wastes test time.

4. Aspheric departure

For aspheric lenses, the SAG of the best-fit sphere serves as the reference. The aspheric departure is the difference between the actual aspheric SAG and the spherical SAG at each zone. Knowing this departure drives the choice of fabrication method, conventional grinding, diamond turning, or molding.

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