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Lensmaker’s Equation Calculator
v2.0
Thin & thick lens analysis • Solve for f, P, n, R1, R2 • Interactive visualization
Uses the standard thin-lens Lensmaker equation (d ≅ 0).
Sign Convention
Light travels left → right. Positive radius = center of curvature to the right (convex surface).
R > 0 (convex)
R < 0 (concave)
R = ∞ (planar)
For R1 (left surface): positive = bulging left, negative = bulging right.
For R2 (right surface): positive = bulging right, negative = bulging left.
For R2 (right surface): positive = bulging right, negative = bulging left.
Refractive index varies with wavelength & temperature. Use manufacturer data for precision.
Relative refractive index nrel = nlens / nmedium.
Ready
Results
Focal Length f
— m
Optical Power P
— D
Lens Type
—
Relative n
—
nlens / nmedium
C1 (1/R1)
— m⁻¹
C2 (1/R2)
— m⁻¹
Interpretation
—
Click Calculate to see the step-by-step derivation.
Lens Visualization
Schematic — rays are illustrative. Light travels left → right.
Parameter Sensitivity
Hover for values. Vertical line = current operating point.
Lens Comparison
| Design | n | R₁ | R₂ | d | f (m) | P (D) | Type |
|---|
Click Add Design to save current settings. Up to 6 designs.
Lensmaker Reference
| Symbol | Quantity | SI Unit | Description |
|---|---|---|---|
| f | Focal length | m | Distance from lens to focal point |
| n | Refractive index | — | Optical index ratio (lens / medium) |
| R₁ | First surface radius | m | Curvature radius of left surface |
| R₂ | Second surface radius | m | Curvature radius of right surface |
| d | Center thickness | m | Lens thickness (thick-lens mode) |
| P | Optical power | D (m⁻¹) | P = 1/f (f in meters) |
| C | Surface curvature | m⁻¹ | C = 1/R |
Refractive Index Reference
| Material | n (approx.) | Notes |
|---|---|---|
| Air | 1.000 | Standard reference |
| Water | 1.333 | At 20°C, 589 nm |
| Fused silica | 1.458 | At 589 nm |
| Crown glass | 1.517 | Typical BK7 |
| Acrylic (PMMA) | 1.490 | Common optical plastic |
| Polycarbonate | 1.585 | High-index plastic |
| Flint glass | 1.620 | Dense flint |
Refractive index varies with wavelength, temperature, material grade, and composition. Manufacturer optical-glass data should be used for precision lens design.
Related Optics Calculations
The Lensmaker equation models the lens itself; the imaging equation relates object and image distances.
Object / Image Analysis
m
Positive = object on left side of lens.
How to Use This Tool
- Select Thin Lens or Thick Lens mode.
- Choose what variable to Solve For (f, P, n, R₁, or R₂).
- Select a lens material preset or enter a custom refractive index.
- Enter R₁ and R₂ with your preferred units.
- Follow the sign convention shown above — positive = convex, negative = concave.
- In Thick Lens mode, enter the center thickness d.
- Select the surrounding medium (air, water, or custom).
- Click Calculate to see results, steps, visualization, and sensitivity chart.
- Use Geometry Presets to quickly explore common lens shapes.
- Add designs to the Comparison table to study different configurations.
What This Tool Offers
Lensmaker equation (thin & thick)
Multi-variable solve (f, P, n, R₁, R₂)
Automatic unit conversion
Optical power & curvature
Geometry presets
Sign convention assistant
Interactive lens visualization
Parameter sensitivity chart
Material presets + custom n
Calculation breakdown
Lens comparison table
Object/image analysis
Reference tables
Engineering interpretation
Copy results
Assumptions & Limitations
- Paraxial approximation — rays are close to the optical axis.
- Spherical surfaces — assumes perfect spherical curvature.
- Homogeneous, isotropic material — uniform refractive index.
- Monochromatic light — single wavelength (dispersion not modeled).
- Thin-lens approximation — lens thickness is negligible (d → 0).
- Thick-lens model — includes center thickness but neglects higher-order effects.
- No aberrations — spherical, chromatic, and other aberrations are not modeled.
- Manufacturing tolerances — real lenses vary from ideal values.
- Environmental effects — temperature, pressure, and humidity are ignored.
This tool is intended for educational and preliminary optical design. For precision multi-element systems, use professional optical-design software.
References & Further Reading
- Hecht, E. (2017). Optics, 5th ed. Pearson.
- Born, M., & Wolf, E. (2019). Principles of Optics, 7th ed. Cambridge.
- Smith, W. J. (2007). Modern Optical Engineering, 4th ed. McGraw-Hill.
- NIST (2023). Refractive Index Database. nist.gov.
- Schott AG. (2023). Optical Glass Data Sheets. schott.com.
All numerical reference data is approximate and intended for educational use.
Vorzep Lensmaker’s Equation Calculator • v2.0 • Built for optical engineering & education
Variables
| Symbol | Quantity | Unit |
|---|---|---|
| f | Focal length | m |
| n | Lens refractive index | |
| R1 | Radius of curvature of first surface | m |
| R2 | Radius of curvature of second surface | m |
What it means
The lensmaker’s equation relates the focal length of a thin lens to its refractive index (n) and the radii of curvature of its surfaces (R₁ and R₂). The sign convention is important. This equation is used by optical designers to calculate lens curvatures for desired focal lengths. It is fundamental for designing camera lenses, eyeglasses, microscope objectives, and many other optical systems. Understanding this equation is essential for lens manufacturing and optical engineering. It assumes the lens is in air and is thin; for thick lenses, a more complex form is used.
Worked example
Lensmaker's Equation – Two Detailed Examples
Real‑World
Scenario: An optical engineer is designing a bi‑convex lens from glass with refractive index n = 1.5. The radii of curvature of the two surfaces are R₁ = 20 cm and R₂ = −20 cm (using sign convention: R₂ is negative for the second surface). They compute the focal length using 1/f = (n−1)(1/R₁ − 1/R₂) = (0.5)(1/20 − 1/(−20)) = 0.5×(1/20 + 1/20) = 0.5×0.1 = 0.05, so f = 20 cm. This gives them the required lens curvature for a 20 cm focal length lens.
| Parameter | Value |
|---|---|
| n | 1.5 |
| R1 (cm) | 20 |
| R2 (cm) | −20 |
11/f = (1.5−1) × (1/20 − 1/(−20)) = 0.5 × (0.05 + 0.05) = 0.5 × 0.1 = 0.05
2f = 1/0.05 = 20 cm
Result
20 cm
✓ Focal length
Scenario: A lens manufacturer needs to create a plano‑convex lens (one side flat, R₂ = ∞) with focal length 30 cm using glass of n = 1.5. They solve for R₁: 1/30 = (1.5−1)(1/R₁ − 0) → 1/30 = 0.5/R₁ → R₁ = 15 cm. They grind the convex surface with a radius of 15 cm to achieve the desired power. This calculation is routine in lens fabrication.
| Parameter | Value |
|---|---|
| n | 1.5 |
| f (cm) | 30 |
| R2 | ∞ |
11/30 = 0.5 × (1/R1 − 0) → 1/30 = 0.5/R1
2R1 = 0.5 × 30 = 15 cm
Result
15 cm
✓ Required radius
Insight: The Lensmaker's equation relates the focal length of a thin lens to its refractive index and surface curvatures. It is the fundamental formula for lens design and manufacturing.
Common mistakes
- Lensmaker’s equation: 1/f = (n−1)·(1/R₁ − 1/R₂) – for a thin lens in air.
- Sign convention for radii: R₁ is the radius of curvature of the first surface (light incidence side). R₂ of the second. Positive if the centre of curvature is on the outgoing side (for light).
- n: Refractive index of the lens material relative to air – must be > 1.
- Units: R₁, R₂, and f in the same units (e.g., metres).
- Assumes: Thin lens in air – for lenses in other media, adjust (n_lens/n_medium − 1).
Applications
The lensmaker's equation, 1/f = (n−1)(1/R₁ − 1/R₂), relates the focal length of a thin lens to its refractive index (n) and the radii of curvature (R₁, R₂) of its two surfaces. This is used by optical engineers to design lenses with specific focal lengths, to select materials, and to optimise lens shapes for aberration control. It is essential in designing lenses for cameras, microscopes, telescopes, and eyeglasses. By choosing appropriate radii and glass types, engineers can achieve the desired optical properties. This equation also underpins the design of complex lens systems. Understanding it is fundamental for anyone involved in lens manufacturing or optical system design.
- Optical lens design and manufacturing
- Selection of glass materials for desired focal lengths
- Design of achromatic and apochromatic lens systems
- Camera lens and microscope objective design
- Educational understanding of lens curvature effects
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