Edmund Optics SAG Calculator: How to Use It

The Edmund Optics SAG calculator is a free online tool that computes the sagitta (sag) of a curved optical surface—the depth of the curve measured from the chord (flat reference plane) to the surface—when you enter the radius of curvature and the clear aperture diameter. You get the sag value in the same units you input, which is essential for lens design, grinding, and polishing. This article explains the formula, how to use the calculator, and common pitfalls.

Formula

The sagitta (s) of a spherical surface is given by:

s = R – √(R² – (D/2)²)

Where:

• s = sagitta (depth of the curve), in mm (or any unit)
• R = radius of curvature of the spherical surface, in mm
• D = clear aperture diameter (the diameter of the surface that actually passes light), in mm

The formula assumes a perfect spherical surface. For aspheric surfaces, the sag is different and requires higher-order terms.

Worked Example

Using the default values from the calculator (R = 100 mm, D = 50 mm):

1. Compute D/2 = 25 mm.
2. Square it: 25² = 625 mm².
3. Subtract from R²: 100² – 625 = 10000 – 625 = 9375 mm².
4. Take the square root: √9375 ≈ 96.8246 mm.
5. Subtract from R: 100 – 96.8246 = 3.1754 mm.

The calculator rounds to 3 decimals, giving s = 3.175 mm. This matches the output.

When to Use the SAG Calculator

Use the sag calculator when you need to know the physical depth of a curved optical surface. This is common in:

• Designing lenses (choosing a radius for a given focal length and aperture).
• Grinding and polishing glass blanks—you need to know how much material to remove.
• Checking if a curved surface will fit in a mechanical mount (depth clearance).
• Estimating the volume of a lens or mirror (for weight or material cost).

The calculator works for both convex and concave surfaces—the sag is the same magnitude; the sign just indicates direction (positive for concave, negative for convex, depending on convention). The Edmund Optics tool itself doesn’t ask for sign; you interpret it.

How to Read the Result

The sag value is the maximum depth of the curve from the flat plane that touches the edge of the aperture. For a concave surface (like a mirror), it’s the depth of the bowl. For a convex surface (like a lens front), it’s the height of the dome. The result is always in the same unit you entered for R and D. If you mix units (e.g., R in mm, D in inches), the result will be wrong—always convert to a single unit first.

The second output, R – s, is just the distance from the center of curvature to the sag point. It’s rarely needed but can help in geometry checks.

Common Mistakes and Misconceptions

• Using the full diameter instead of the clear aperture: The formula uses the clear aperture, not the physical diameter of the lens blank. If you use the blank diameter, you’ll overestimate the sag.
• Unit mix-ups: Always use the same units for R and D. If R is in mm and D is in cm, convert D to mm (or R to cm) before calculating.
• Confusing sag with focal length: The sag is purely geometric; it doesn’t directly give focal length. Focal length depends on refractive index (for lenses) or is R/2 for mirrors.
• Assuming it works for aspherics: The formula is only for spherical surfaces. For parabolic or other aspheric shapes, the sag is different and requires conic constants or polynomial terms.

Limitations of the Formula

The spherical sag formula is exact for a perfect sphere, but it has practical limits:

• If the aperture diameter is larger than twice the radius of curvature (D > 2R), the square root becomes imaginary—the surface would be more than a hemisphere. In practice, optical surfaces are rarely that deep, but be aware.
• For very steep curves (D close to 2R), the sag becomes large and the formula still works, but manufacturing such a surface is challenging.
• The formula ignores edge thickness and mounting chamfers. For a real lens, the physical thickness includes the edge thickness plus the sag.

For most optical design work, the spherical approximation is sufficient. For precision work, use optical design software that handles aspherics and exact ray tracing.

Alternatives and Related Calculations

If you need to calculate other lens parameters, there are related formulas:

• Focal length of a mirror: f = R/2 (for a spherical mirror).
• Lensmaker’s equation: 1/f = (n-1)[1/R1 – 1/R2 + (n-1)t/(n R1 R2)] for thick lenses.
• Sag for a parabolic mirror: s = D² / (16f) where f is the focal length.

Many online calculators combine these, but the Edmund Optics sag calculator focuses on the single geometric quantity. For a quick check, you can also use the approximation s ≈ D²/(8R) for shallow curves (D << R), which is accurate to about 1% when D < 0.5R.

Frequently Asked Questions

What does “sag” mean in optics?

Sag (sagitta) is the depth of a curved optical surface—the distance from the flat plane that touches the edge of the aperture to the deepest (or highest) point of the surface. It’s a geometric property used in lens design and fabrication.

Can I use this calculator for mirrors as well as lenses?

Yes. The sag formula is purely geometric and applies to any spherical surface, whether it’s a mirror, lens, or window. For mirrors, the sag is often used to determine the depth of the concave surface for grinding.

What if I don’t know the radius of curvature?

You need the radius of curvature to compute sag. If you don’t have it, you can measure it with a spherometer, or derive it from the focal length (for a mirror, R = 2f). For a lens, you’d need the lensmaker’s equation and the refractive index.

Does the calculator work for convex surfaces?

Yes, the magnitude of the sag is the same for a convex surface with the same radius and aperture. The sign (positive or negative) is a convention; the calculator gives a positive value, which you can interpret as depth for concave or height for convex.

Why does my result differ from a spherometer reading?

Spherometers measure sag using a ring of a specific diameter, not the full clear aperture. If the spherometer’s ring diameter is smaller than the clear aperture, the sag will be different. Also, spherometers have their own calibration and may include a correction factor for the ball contact points.

Is the sag the same as the thickness of the lens?

No. The sag is only the depth of the curve. The total thickness of a lens is the edge thickness plus the sag (for a plano-convex lens) or the sum of sags plus center thickness for biconvex lenses. The calculator only gives the sag, not the total thickness.

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in r | Radius of curvature (R) | mm | 100 | 0
in d | Clear aperture diameter (D) | mm | 50 | 0
out sag | Sagitta (s) | mm | r - sqrt(r^2 - (d/2)^2) | 3
out r_minus_sag | r - s (for reference) | mm | r - (r - sqrt(r^2 - (d/2)^2)) | 3