What is the Lensmaker's equation?

1/f = (n−1)[1/R₁ − 1/R₂], where f is focal length, n is refractive index, and R₁, R₂ are the radii of curvature of the lens surfaces. This thin-lens form assumes the lens is in air and its thickness is negligible compared to focal length and object distances.

What is optical power?

Optical power P = 1/f measured in diopters (D) when f is in meters. It indicates the degree to which a lens converges (P > 0) or diverges (P < 0) light. Used universally in ophthalmology — eyeglass prescriptions, contact lenses, IOL implants are all specified in diopters.

How does refractive index affect focal length?

For the same surface curvatures, higher index → shorter focal length (more power). A glass lens with n=1.5 has half the power of one with n=2.0 for identical R₁ and R₂. This is why high-index eyeglass lenses can be thinner: same prescription requires less surface curvature.

What is the sign convention for radii?

For light traveling left → right: R > 0 means the center of curvature is on the right side of the surface (incoming light hits a convex surface). R 0 (left surface convex), R₂ < 0 (right surface convex when viewed from incoming side).

What's the difference between a thin and thick lens?

A "thin lens" has thickness much smaller than its focal length and object distances. The thin Lensmaker's equation is accurate for typical eyeglasses and most camera elements. Thick lenses (thickness comparable to f) need the full equation: 1/f = (n−1)[1/R₁ − 1/R₂ + (n−1)d/(nR₁R₂)], where d is lens thickness.

How do I design a +2.00 D reading lens?

You need f = 1/2.00 m = 50 cm. Pick a material (CR-39, n = 1.498). Decide on shape (planoconvex easiest): R₂ = ∞. Solve: R₁ = (n−1) × f = 0.498 × 50 = 24.9 cm. Grind a 25 cm radius convex surface on the front, leave the back flat. Power = +2.00 D, suitable for moderate reading.

Why do achromat doublets pair crown and flint glass?

Crown glass (n=1.5) has low chromatic dispersion (high Abbe number); flint glass (n=1.7) has high dispersion. Pairing a converging crown with a slightly diverging flint balances chromatic dispersion across both colors while maintaining net converging power. The combination cancels chromatic aberration at two specific wavelengths, much better than either element alone.

Lensmaker's Equation Calculator

Q: How does refractive index affect focal length?

For the same surface curvatures, higher index → shorter focal length (more power). A glass lens with n=1.5 has half the power of one with n=2.0 for identical R₁ and R₂. This is why high-index eyeglass lenses can be thinner: same prescription requires less surface curvature.

Q: What is the sign convention for radii?

For light traveling left → right: R > 0 means the center of curvature is on the right side of the surface (incoming light hits a convex surface). R 0 (left surface convex), R₂ < 0 (right surface convex when viewed from incoming side).

Q: What's the difference between a thin and thick lens?

A "thin lens" has thickness much smaller than its focal length and object distances. The thin Lensmaker's equation is accurate for typical eyeglasses and most camera elements. Thick lenses (thickness comparable to f) need the full equation: 1/f = (n−1)[1/R₁ − 1/R₂ + (n−1)d/(nR₁R₂)], where d is lens thickness.

Q: How do I design a +2.00 D reading lens?

You need f = 1/2.00 m = 50 cm. Pick a material (CR-39, n = 1.498). Decide on shape (planoconvex easiest): R₂ = ∞. Solve: R₁ = (n−1) × f = 0.498 × 50 = 24.9 cm. Grind a 25 cm radius convex surface on the front, leave the back flat. Power = +2.00 D, suitable for moderate reading.

Q: Why do achromat doublets pair crown and flint glass?

Crown glass (n=1.5) has low chromatic dispersion (high Abbe number); flint glass (n=1.7) has high dispersion. Pairing a converging crown with a slightly diverging flint balances chromatic dispersion across both colors while maintaining net converging power. The combination cancels chromatic aberration at two specific wavelengths, much better than either element alone.

Use the Lensmaker's equation to determine the focal length and optical power (in diopters) of a lens from its refractive index and radii of curvature.

The Lensmaker's equation is what lens designers use to specify the actual physical shape of a lens. The thin-lens equation tells you what focal length you need; the Lensmaker's equation tells you how to grind the glass to achieve it. Given the refractive index of the material (1.5 for standard crown glass, 1.78 for high-index ophthalmic glass, 1.5 for plastic CR-39, or 2.0+ for some specialized glasses), and the two surface radii of curvature, the equation 1/f = (n−1)[1/R₁ − 1/R₂] gives the focal length.

This calculator handles the standard "thin lens in air" case. Enter n and the two radii, and you get f and the optical power (P = 1/f in diopters when f is in meters — the standard ophthalmic unit). Eyeglasses, contact lenses, and intraocular implants are all specified by their optical power in diopters, calculated this way. Camera and microscope objectives use the same formula for each individual element, then combine elements to achieve final system focal length and correct aberrations.

The sign convention matters and trips up new students. R₁ and R₂ are measured from the lens center outward to the center of curvature of each surface. Convex surfaces facing the incoming light have positive R; concave surfaces facing incoming light have negative R. A symmetric biconvex lens has R₁ > 0 (front convex) and R₂ < 0 (back convex when viewed from the rear, but signed negative because center of curvature is to the left of the lens). Memorize this and the formula gives sensible answers; get it wrong and you'll predict a converging lens that diverges.

Inputs

Refractive Index n

Radius of Curvature R₁ (cm)

Positive = center of curvature on right side

Radius of Curvature R₂ (cm)

Negative = center of curvature on left side

Results

Focal Length

20.00 cm

Optical Power

5.00 D

Type

Converging (positive)

Lensmaker's Equation Results

Parameter	Value
Focal Length f	20.0000 cm
Optical Power P	5.0000 D (diopters)
Refractive Index n	1.5000
R₁	20 cm
R₂	-20 cm
Lens Type	Converging (positive)
(n-1)	0.5000
Formula	1/f = (n−1)[1/R₁ − 1/R₂]

Last updated: May 28, 2026

Formula

**Lensmaker's equation (thin lens in air):** 1/f = (n − 1) × [1/R₁ − 1/R₂] Where: - **f**: focal length (positive = converging, negative = diverging) - **n**: refractive index of lens material - **R₁**: radius of curvature of front surface (light hits this first) - **R₂**: radius of curvature of back surface **Sign convention for radii:** Measure each R from the surface vertex (where the optical axis crosses the surface) to the center of curvature of that surface: - **R > 0**: center of curvature is on the same side as outgoing (transmitted) light. - **R < 0**: center of curvature is on the same side as incoming light. - **R = ∞ (or "flat")**: 1/R = 0; surface is plano. For light traveling left → right: - Biconvex (both convex): R₁ > 0, R₂ < 0 - Biconcave (both concave): R₁ < 0, R₂ > 0 - Plano-convex (flat front, convex back): R₁ = ∞, R₂ < 0 - Plano-concave (flat front, concave back): R₁ = ∞, R₂ > 0 - Meniscus convex (convex front, less curved convex back): R₁ > 0, R₂ > 0 (both on right of surface) **Optical power (diopters):** P = 1/f (with f in meters) P = +1 D → f = 1 m (mild reading glasses) P = +2.50 D → f = 0.40 m (moderate reading glasses) P = −2.00 D → f = −0.50 m (mild myopia correction) P = −6.00 D → f = −0.167 m (moderate myopia correction) **Worked example: symmetric biconvex glass lens** n = 1.50, R₁ = +20 cm, R₂ = −20 cm. 1/f = (1.5 − 1) × [1/20 − 1/(−20)] = 0.5 × [0.05 + 0.05] = 0.5 × 0.1 = 0.05 cm⁻¹ f = 20 cm. Power = 100/20 = 5.0 D. **Worked example: high-index ophthalmic lens** Same shape but n = 1.74 (high-index glass for eyeglasses): 1/f = (1.74 − 1) × 0.1 = 0.74 × 0.1 = 0.074 cm⁻¹ f = 13.5 cm. Same physical curvature gives shorter focal length (more power) with higher index — why high-index glasses can be thinner. **Common lens materials and refractive indices:** | Material | n | Use | |---|---|---| | Crown glass (BK7) | 1.517 | Standard achromat element | | Flint glass (SF11) | 1.78 | Achromat doublet correction | | CR-39 (plastic) | 1.498 | Standard plastic eyeglasses | | Polycarbonate | 1.586 | Impact-resistant glasses, safety | | Trivex | 1.532 | Lightweight plastic | | 1.67 index plastic | 1.67 | High-index plastic eyeglasses | | 1.74 index glass/plastic | 1.74 | Thinnest high-power lenses | **Power-thickness trade-off in eyeglasses:** Higher index → same power with less material curvature → thinner lenses. A −6 D myopia correction: - n = 1.50 plastic: ~5 mm edge thickness - n = 1.74 high-index: ~3 mm edge thickness That's the marketing reason high-index lenses exist.

How to use this calculator

Pick the refractive index n of your lens material (1.5 for standard glass/plastic, higher for thinner lenses).
Enter the radius of curvature R₁ for the front surface and R₂ for the back.
Use the sign convention: positive R = center of curvature on the right side of the surface.
For a flat surface, set R to a very large number (e.g., 1000000 effectively makes 1/R = 0).
Output: focal length f and optical power P. Power in diopters is f in meters inverted.
For eyeglasses, optical power is the prescription number (e.g., −2.50 D myopia correction).

Worked examples

Reading glasses for +2.50 D prescription

**Scenario:** Design a +2.50 D reading glass lens using CR-39 plastic (n = 1.498). Use a planoconvex shape with R₂ = ∞ (flat back). **Calculation:** f = 1/2.50 D = 0.40 m = 40 cm. Lensmaker: 1/40 = (1.498−1) × [1/R₁ − 0] = 0.498/R₁. Solve: R₁ = 0.498 × 40 = 19.92 cm ≈ 20 cm. **Result:** A planoconvex CR-39 lens with R₁ = 20 cm (front) and R₂ = ∞ (flat back) gives +2.50 D power. Switch to high-index n = 1.67: R₁ = 0.67 × 40 = 26.8 cm — less curved, thinner edge. Material choice affects appearance even at the same prescription.

Camera lens single element

**Scenario:** A symmetric biconvex glass element (n = 1.52) with both radii 50 mm magnitude. What's the focal length? **Calculation:** R₁ = +50 mm, R₂ = −50 mm. 1/f = (1.52 − 1) × [1/50 − 1/(−50)] = 0.52 × [0.02 + 0.02] = 0.52 × 0.04 = 0.0208 mm⁻¹. f = 48 mm. **Result:** A 48 mm focal length single element. Real 50mm camera lenses use multiple elements (often 6–8) to correct aberrations — chromatic, spherical, coma, distortion. Each element's power adds approximately: 1/f_total ≈ Σ(1/f_i). The Lensmaker's equation designs each one; the total is then optimized via ray-tracing.

Comparing high-index ophthalmic lenses

**Scenario:** Design a −4.00 D myopia-correction lens. Compare standard (n=1.50) and high-index (n=1.74) options, both biconcave with R₁ = −R₂. **Calculation:** Need f = 1/(−4.00) = −0.25 m = −25 cm. n=1.50 case: 1/(−25) = 0.5 × [1/R₁ − 1/R₂] = 0.5 × 2/R₁ = 1/R₁ → R₁ = −25 cm (so R₁ = −25, R₂ = +25). n=1.74 case: 1/(−25) = 0.74 × 2/R₁ = 1.48/R₁ → R₁ = −37 cm. **Result:** High-index lens (n=1.74) needs gentler curves (37 cm vs 25 cm radius) to achieve same prescription. This translates to thinner lenses, lighter weight, less "Coke bottle" thickness — primary appeal for high-prescription wearers. Cost is higher index materials are more expensive and have more chromatic aberration (lower Abbe number).

When to use this calculator

**Use the Lensmaker's equation for:**

- **Ophthalmic lens design**: prescription glasses, contact lenses, IOL (intraocular lens) implants. - **Camera and microscope lens element design**: each individual glass element's focal length. - **Optical bench experimentation**: deriving the shape of a needed test lens. - **Educational and homework problems**: understanding how lens shape determines focal length. - **Material selection**: comparing same shape at different index, or same f at different materials. - **Thin-film optics simulations**: using the equation as a baseline before adding coatings.

**Optical power (diopters) is the dominant unit in ophthalmology:**

- Eyeglass prescriptions: typical range −20 D (severe myopia) to +10 D (severe hyperopia). - Reading glasses ("add"): typically +1.00 to +3.50 D added to distance prescription. - Contact lenses: same range but specified differently (curvature also given). - Standard cataract IOLs: 12–25 D depending on eye axial length. - Camera lenses: rarely expressed in diopters; focal length is the standard.

**Lens shapes and applications:**

| Shape | R₁, R₂ signs | Use | |---|---|---| | Biconvex (symmetric) | + / − | General converging lens, basic +D power | | Biconcave (symmetric) | − / + | Basic diverging lens, basic −D power | | Plano-convex | ∞ / − | Easier manufacture, less spherical aberration | | Plano-concave | ∞ / + | Diverging, easier to manufacture | | Positive meniscus | + / + (both right) | Concentric design, less coma | | Negative meniscus | − / − (both left) | Mostly used in eyeglasses |

**When the Lensmaker's equation isn't enough:**

- **Thick lenses**: when lens thickness is not negligible vs focal length. Use the full Lensmaker's equation with thickness term: 1/f = (n−1)[1/R₁ − 1/R₂ + (n−1)d/(nR₁R₂)]. - **Multi-element systems**: combine element powers approximately (1/f_total ≈ Σ 1/f_i for closely-spaced elements), then ray-trace for precision. - **Aspheric lenses**: spherical surfaces have aberrations. Aspheric (parabolic, hyperbolic, conic) surfaces eliminate them at one wavelength but the Lensmaker's equation no longer applies. - **Chromatic correction**: at multiple wavelengths, n varies. Achromat doublets pair high-index/low-Abbe flint with low-index/high-Abbe crown.

**Achromat design rough rule:**

To zero net chromatic dispersion with two elements in contact:

P_crown × V_crown = −P_flint × V_flint

Where V is the Abbe number (a measure of low chromatic dispersion; crown glass V ≈ 60, flint V ≈ 25). The result: pair a converging crown (high V) with a diverging flint (low V) to net positive power but neutral color.

Common mistakes to avoid

Confusing the radius of curvature sign convention. The conventions vary by textbook; use the one consistent with the Lensmaker formula being used.
Forgetting that R is to the center of curvature, not the edge or the diameter. R is the sphere's radius, which can be far from the lens's physical extent.
Mixing units of R and f. Both must be in the same units (typically cm or mm).
Trying to use the Lensmaker for thick lenses without the thickness correction. For thick lenses (lens d > f/10), use the full version.
Calculating power in diopters with f in cm. Diopters = 1/f only when f is in meters. f = 20 cm → P = 5 D (not 0.05 D).
Forgetting refractive index varies with wavelength. The Lensmaker's gives f at one wavelength; chromatic aberration in single-element lenses means different colors focus at different points.
Confusing optical power with magnification. Power describes the lens itself; magnification depends on object distance as well.

Frequently Asked Questions

Sources & further reading

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