Lensmaker's equation

When the same singlet sits in water, oil, or any other medium, that one is replaced by the ratio of the two indices: one over f equals n_lens over n_medium minus one, times the same curvature difference. This capability evaluates that closed form and returns the focal length in the same length unit as the radii. A surface radius of exactly zero would divide by zero and is rejected as invalid_input.

What the lensmaker equation assumes and when it applies

The focal length is the reciprocal of that power. Radii follow the Cartesian sign convention with light traveling left to right: a radius is positive when the centre of curvature lies to the right of the surface. Real thick lenses need the extra centre-thickness term; aspheres, gradient-index glass, and multi-element objectives need ray tracing. This tool returns only the ideal thin-lens focal length so you can check textbook problems, seed a design, or document a baseline before detailed modeling. The thin-lens lensmaker equation is the first-order model taught in introductory optics: the lens thickness is negligible compared with the radii and with the object and image distances, both faces are spherical, and the surrounding medium is homogeneous with refractive index n_medium. Under those assumptions the optical power of the two surfaces adds, and the combined power in the surrounding medium is n_lens over n_medium minus one, times the difference of the surface curvatures. In water n_medium is about 1.333, so a glass singlet that is strong in air becomes much weaker because the index contrast shrinks. A surface radius of zero is not accepted as a planar face here: the formula would divide by zero, so the capability fails with invalid_input instead of inventing an infinite radius.

Input contract, response shape, and error behaviour

Both refractive indices must be strictly positive. Convert millimetres to metres before calling if your drawings use millimetres; mixing units produces a well-formed number that is wrong for the geometry you intended. Each field may arrive as a JSON number or as a numeric string. Missing fields, empty strings, and non-finite values fail with invalid_input naming the field, and those failures are not billed. Equal indices n_lens equals n_medium produce zero optical power and an undefined focal length, so that case is also invalid_input. On success the body echoes the four inputs, reports focal_length in the same length unit as the radii, and includes a formula string for audit logs. Pricing for a successful analytic call is a flat $0.002 per item.

Where a thin-lens focal length API earns its keep

Product teams sizing simple magnifiers, LED collimators, or phone-camera singlet concepts need a transparent power estimate before they open a full optical design package, especially when the singlet will sit behind a window or in a fluid rather than in air. Instrumentation groups compare measured focal lengths against the thin-lens prediction to flag wrong glass type, inverted surface order, or a setup that is no longer thin. Failed validation does not consume balance. Keep classroom centimetre radii inside ordinary SI magnitudes, pin the rounded focal length in tests, and you get reproducible first-order optics for homework, LMS plugins, and edge workers without standing up a ray tracer of your own. Physics and engineering courses use the calculator to verify homework and lab prep: students pick crown glass near n_lens 1.5, choose a surrounding medium of air or water, pick symmetric biconvex radii, and confirm that f matches the expected value before grinding or measuring on an optical bench. Underwater imaging, oil-immersion microscope objectives, and index-matching baths are the cases where n_medium is not one and the air-only formula would lie; sending the true surrounding index is the whole point of this landing. Because each successful call is a single analytic item at $0.002, scripts can sweep families of radii or index tables without a subscription or model cost. Prefer this endpoint when you want a stable, reviewable closed form with an explicit surrounding medium, not a thick-lens solver, not multi-element ray tracing, and not wavelength-dependent dispersion, which require additional thickness, spacing, and glass catalog inputs beyond this contract.