Physics and engineering
Mirror Conversion Calculator
Convert a spherical mirror specification among signed radius, focal length, optical power, curvature, f-number, and exact edge sag under one declared sign convention.
CURRENT MODEL
Choose the known mirror specification and aperture
Optics students, telescope makers, inspection technicians, and buyers translating a spherical mirror drawing or catalog specification before layout or fabrication.
Decision supportedDetermine whether radius, focal length, optical power, clear aperture, and spherical sag describe the same concave or convex surface.
LIVE SPHERICAL SECTION
Signed mirror geometry
The live section changes curvature direction, focal-side marker, aperture chord, and sag annotation from the current mirror type and dimensions.
| Quantity | Symbol or equation | Current value | Unit |
|---|
How to use
Translate one drawing value without losing sign or units
- Select concave or convex before entering a magnitude; that choice establishes the sign convention.
- Identify whether the known number is radius in millimetres, focal length in millimetres, or optical power in diopters.
- Enter the positive known magnitude rather than manually adding a negative sign for a convex mirror.
- Enter the actual clear aperture, not the outside diameter of the mirror blank or mount.
- Compare signed focal length and power with the geometric sag and f-number before transferring values to a drawing.
- Save the exact ledger with the surface identifier, coating side, datum, and measurement uncertainty.
Spherical-mirror fundamentals
Six distinctions behind a clean specification
- Radius sign
- Concave and convex surfaces can share a radius magnitude but focus on opposite signed sides.
- Paraxial focus
- Focal length equals half the radius only for rays sufficiently close to the optical axis.
- Optical power
- Power is reciprocal focal length in metres, so millimetres must be converted before inversion.
- Clear aperture
- The usable illuminated diameter may be smaller than the substrate or coating diameter.
- F-number
- The ratio |f|/D describes cone speed but does not alone establish image quality.
- Spherical sag
- Sag is a surface depth from the vertex plane, not focal distance or coating thickness.
Calculation method
Normalize radius first, then derive reciprocal and aperture quantities
The selected source value is converted to an unsigned radius magnitude. The mirror type then applies one sign to both radius and focal length. This prevents an unsigned catalog radius from being accidentally combined with a signed optical power.
F-number uses the absolute focal length because it describes geometry rather than convergence sign. Sag uses the sphere radius and aperture chord exactly; the square-root domain is checked before any result is shown.
Manufacturing datum
Radius metrology may reference a fitted sphere while sag is measured from a mechanical edge plane. Record both datums before comparing values.
Marginal-ray departure
A low f-number exposes stronger spherical aberration. The conversion is still geometrically correct, but the paraxial focus does not predict every ray intercept.
Power tolerance asymmetry
Because power is reciprocal focal length, symmetric millimetre tolerances become asymmetric diopter errors. Convert tolerance endpoints, not just the nominal value.
Detailed calculation process
Symbols, current substitution, intermediate quantities, and reconciliation
f = R/2; P = 1/f_m; C = 1/R_m; N = |f|/D; s = |R| - sqrt(R^2 - (D/2)^2)All conversions and sag calculations retain full precision. Signs and aperture validity are evaluated before display rounding.
| Symbol | Meaning | Default | Unit |
|---|---|---|---|
| R | Signed spherical radius | +200 | mm |
| f | Signed paraxial focal length | +100 derived | mm |
| P | Reciprocal optical power | 10 derived | D |
| D | Clear aperture diameter | 25 | mm |
| N | Working f-number | 4 derived | dimensionless |
| s | Spherical edge sag | 0.391007 derived | mm |
Waiting for valid inputs.
Interpretation
Use signed optics and unsigned fabrication geometry together
A negative focal length or power identifies a diverging convex mirror; it does not imply a negative physical radius magnitude or sag depth. The f-number can compare cone speed between concave and convex specifications, while edge sag indicates whether the chosen aperture is practical on the declared sphere.
Evidence and measurement
Preserve the surface definition that made the conversion meaningful
Record part number, mirror type, radius source and tolerance, wavelength if focal power was measured, clear-aperture definition, vertex and edge datums, coating side, temperature, instrument calibration, and whether radius came from interferometry, spherometry, or a drawing.
Scope and limitations
What a radius conversion cannot certify
- Conic, aspheric, toroidal, freeform, or segmented surface sag
- Spherical aberration, coma, astigmatism, scatter, or coating phase
- Substrate edge thickness, mount interference, or bevel allowance
- Radius tolerance, power uncertainty, or temperature sensitivity
- Usable aperture after masking, obscuration, or incidence angle
- Optical safety or finished-system image performance
A rotationally symmetric spherical mirror in the paraxial focal-length convention. Concave radius and focal length are positive; convex values are negative. Edge sag is geometric and uses the unsigned sphere radius.
Key terminology
Mirror specification glossary
- Vertex
- Point where the optical axis meets the mirror surface.
- Center of curvature
- Center of the sphere from which the mirror patch is cut.
- Paraxial focal length
- First-order focus distance for rays close to the axis.
- Diopter
- Reciprocal metre unit of optical power.
- Clear aperture
- Diameter guaranteed to transmit or reflect the intended beam.
- Sagitta
- Axial depth of a curved surface relative to its aperture chord plane.
Practical cases
Two conversions with different engineering consequences
Concave alignment mirror
A 200 mm radius and 25 mm clear aperture yield a 100 mm paraxial focal length, 10 D power, f/4 geometry, and 0.391007 mm edge sag. The values form a coherent starting drawing.
Convex inspection substitute
A supplier offers a convex mirror by negative power while the assembly drawing lists radius magnitude. Converting both under one sign convention exposes whether the substitute diverges with the intended focal scale.
Important note
Specification equivalence is not optical acceptance
Use this page to reconcile nominal first-order quantities. Approve a real mirror only with surface figure, roughness, coating, aperture, environmental, and system-level tolerance evidence.
Frequently asked questions
Why does a spherical mirror have f = R/2?
For paraxial rays on a spherical mirror, the small-angle geometry places the focus halfway between the vertex and center of curvature. Marginal rays do not share exactly the same focus.
Why is convex mirror power negative?
This page uses the real-positive convention: a convex mirror has a virtual focus behind the surface, so signed focal length and reciprocal optical power are negative.
Is mirror optical power measured in diopters?
Yes when focal length is expressed in metres, P = 1/f has unit reciprocal metre, conventionally called the diopter.
Does edge sag include coating thickness?
No. Sag is the ideal spherical substrate geometry from vertex plane to the spherical edge at the entered clear aperture.
Why can aperture not exceed twice the radius?
The sag equation intersects a sphere. Beyond the sphere diameter the square-root radicand is negative, so no real spherical surface point exists.
Can I use this for a parabolic telescope mirror?
Only for the paraxial focal relation near the vertex. Parabolic sag differs from spherical sag and requires a conic-surface equation.
Authority and follow-on work
Reliable sources and related calculators
- OpenStax University Physics — Spherical MirrorsSupports spherical-mirror geometry, ray behavior, and the paraxial boundary.
- OpenStax University Physics — Optics Key EquationsLists f = R/2, mirror equation, and magnification conventions.
- NIST Guide to the SISupports metre-based reciprocal units and traceable unit conversion.
Related calculators
Continue with a distinct physics question without silently changing the model boundary.