Diopter Calculator - Lens Power, Stack, Prescription

A diopter calculator converts between the focal length and the optical power of a thin lens, adds the powers of two lenses stacked in contact, and decodes the sphere, cylinder, and axis from a glasses prescription. The diopter calculator is built for first-year physics homework, optometry training, and prescription literacy, where the same reciprocal-metre formula 1/f = P shows up across thin-lens problems, lensmaker's equation, and contact lens vertex compensation.

• • Physics homework: Convert a 200 mm focal length to the matching 5 D power, or go the other direction from a +2.50 D reading add to focal length.
• • Stacked optics: Add the powers of two thin lenses in contact to find the equivalent single lens, such as +2.50 D with -1.25 D.
• • Prescription reading: Translate a glasses prescription of -2.00 -0.75 x 90 into sphere, cylinder, axis, and second-meridian power.
• • Eye anatomy context: Compare a calculated lens power to the roughly 60 D of total optical power of the adult human eye.

The diopter is the SI-derived unit of optical power: 1 D equals the reciprocal of 1 metre of focal length, so a thin converging lens with f = 0.5 m has P = 2 D, and a thin diverging lens with f = -0.25 m has P = -4 D. Reading glasses, contact lens prescriptions, and magnifiers all use this convention.

Once you know the diopter answer for a thin lens, the same reciprocal-length algebra shows up in the surface-radius derivation in the Lensmakers Equation Calculator, which converts refractive index and curvature into focal length and optical power for thick lenses.

The calculator evaluates three independent expressions in one pure function: the reciprocal focal length formula P = 1 / f, the in-contact thin-lens combination P_total = P_1 + P_2, and the prescription second-meridian cumulative power P_2nd = SPH + CYL. Each expression uses diopters as the common unit so the answers can be compared directly.

• f (focal length): Distance from the lens to its focal point, in metres for the formula or mm/cm for the visible input.
• P (optical power): Reciprocal of focal length in metres. 1 D = 1 / m, so a 200 mm converging lens has P = 5 D.
• P_1, P_2 (stacked lens powers): Optical powers of two thin lenses in contact. The combined power is the algebraic sum of the two diopter values.
• SPH (sphere): Spherical power of a glasses prescription, in diopters. Negative for myopia, positive for hyperopia.
• CYL (cylinder): Cylindrical power in diopters, with axis telling which meridian the cylinder acts on.
• Axis: Orientation of the cylinder meridian in degrees, between 0 and 180 per ANSI Z80.1 prescription notation.

Each input feeds one of the three expressions inside the pure function, and the result panel reports optical power, focal length, stacked total power, and the prescription second meridian. The visible numeric answers use two-decimal precision to match the 0.25 D step used in optometry prescriptions.

According to Wikipedia - Dioptre, optical power P in diopters equals one divided by the focal length f in metres, so a 0.5 m converging lens has P = 1 / 0.5 m = 2 D.

According to American Optometric Association - Reading a Prescription, a glasses prescription lists sphere (SPH), cylinder (CYL), and axis, where the second-meridian cumulative power is the sphere plus the cylinder and the axis is the cylinder meridian in degrees.

The diopter reciprocal 1/f in metres is the same form that drives the image-distance equation 1/o + 1/i = 1/f, so the Thin Lens Equation Calculator covers the object and image side of the same thin-lens geometry.

Four ideas explain every diopter calculation: the reciprocal-length definition, the sign convention, the thin-lens combination rule, and the sphere/cylinder/axis prescription grammar.

These four ideas cover every diopter problem in an introductory physics or optometry course. The reciprocal-length definition makes the algebra fast, the sign convention gives every answer a clear meaning, the combination rule treats a stack as one lens, and the prescription grammar turns printed numbers into the matching focal length.

If you already know the diopter power from this calculator, the Focal Length Calculator converts that value into focal length in millimetres for camera, telescope, and projector optics without re-entering the data.

Choose any combination of focal length, optical power, stacked lenses, or prescription values; the result panel updates in real time as you change any field. The Reset button restores the default magnifier and prescription example.

1. 1 Pick a focal length or an optical power: Enter the focal length in mm, cm, or m, or enter the optical power in diopters. Leave one of the two at 0 to use the other.
2. 2 Pick a focal length unit: Choose mm for magnifier math, cm for projector or camera lenses, or m for long-focus optics.
3. 3 Enter the stacked lens pair: Type the diopter values for the first and second thin lens. The combination assumes the two lenses are in contact.
4. 4 Enter the prescription: Type sphere, cylinder, and axis from a glasses prescription. The second-meridian power is sphere plus cylinder.
5. 5 Read the outputs: Optical power, focal length in millimetres, stacked total power, and second-meridian power appear in the result panel.
6. 6 Compare with a textbook example: Try 200 mm focal length, +2.50 D and -1.25 D stacked lenses, or -2.00 -0.75 x 90 to confirm the panel matches a known lens.

For a 200 mm converging magnifier, enter focal length 200 with unit mm and leave power at 0. The result panel should show 5.00 D of optical power and 200.00 mm of focal length. For a -2.00 -0.75 x 90 prescription, the result panel should show sphere -2.00 D, cylinder -0.75 D, axis 90 degrees, and second-meridian cumulative power -2.75 D.

When the printed sphere or cylinder is stronger than about plus or minus 4 D, glasses and contact lens powers no longer match because of the vertex distance, and the Contact Lens Vertex Calculator applies vertex compensation to the same sphere and cylinder inputs.

The diopter calculator delivers a fast, reciprocal-length answer for every thin-lens and prescription problem a physics student or optometry trainee has to solve.

• • Three calculations in one panel: Reports focal length to optical power, power to focal length, stacked lens total, and prescription second meridian.
• • Two-decimal precision: Rounds every diopter answer to two decimals, matching the 0.25 D step used in optometry prescriptions and contact lens fitting.
• • Unit toggle included: Accepts focal length in millimetres, centimetres, or metres so magnifier, camera, and projector optics stay on the same form.
• • Built-in range checks: Rejects zero focal length and zero optical power because 1/0 is not physically meaningful for a thin lens.
• • Eye anatomy reference: Pairs every answer with the roughly 60 D total optical power of the human eye as a sanity check.

For first-year physics problems the diopter calculator gives the answer in seconds, and the prescription reader turns a printed glasses prescription into the cumulative power on each meridian.

Three input factors change every output, and the calculator surfaces a few approximation caveats worth knowing before quoting a result in a report.

• • The thin-lens formula ignores thickness. Real spectacle lenses have measurable thickness, and the lensmaker's equation is needed when thickness matters.
• • The in-contact combination assumes zero spacing between lenses. Real stacked optics with air gaps need the more general Gaussian lens formula.
• • The prescription reader assumes a single-vision prescription. Bifocal and progressive addition lenses use a separate ADD value that this calculator does not model.

For a full prescription workflow, pair the diopter calculator with the contact-lens vertex calculator for vertex compensation, the focal length calculator for the matching focal length in millimetres, and the lensmaker's equation for thick-lens refractive index math.

As published by Wikipedia - Eye, As published in the Wikipedia article on the eye, the adult human eye has roughly 60 diopters of total optical power, with the cornea providing about 43 D and the crystalline lens about 19 D.

Curved mirrors use the same reciprocal-metre diopter relation for their focal length, so the Mirror Equation Calculator is the natural companion for relating optical power to image distance in a mirror system.