Radius of Curvature Calculator (R₂)
Use our free radius of curvature calculator to determine the second surface radius (R₂) of any lens. This radius of curvature calculator uses the rearranged lens maker's equation to compute R₂ from known focal length, refractive index, and first surface radius. The complete radius of curvature calculator for optical engineers and lens designers.
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1. Radius of Curvature Calculator Tool
Our radius of curvature calculator instantly computes R₂ from your lens parameters. Enter the focal length, refractive index, and R₁, and the radius of curvature calculator will determine the required second surface radius for your lens design.
Radius of Curvature Calculator (R₂)
Enter focal length (f), refractive index (n), and R₁ to calculate R₂
Formula
1/f = (n-1)(1/R₁ - 1/R₂)
Typical: 1.5 (glass), 1.33 (water), 1.52 (crown glass)
Positive for convex, negative for concave
Positive for converging, negative for diverging
2. Understanding R₂ in Lens Design
R₂ is the radius of curvature of the second (back) surface of a lens. The radius of curvature calculator helps determine this critical parameter when designing lenses. Understanding how R₂ affects lens performance is essential for using the radius of curvature calculator effectively.
The second surface radius works together with R₁ to determine the overall focal length. When using the radius of curvature calculator for R₂, remember that the sign of R₂ is typically opposite to R₁ for converging lenses. The radius of curvature calculator handles all sign conventions automatically.
Biconvex Lens
R₁ > 0, R₂ < 0: The radius of curvature calculator typically gives negative R₂ for the back surface of converging lenses.
Biconcave Lens
R₁ < 0, R₂ > 0: The radius of curvature calculator gives positive R₂ for diverging lens back surfaces.
3. The R₂ Formula
The radius of curvature calculator uses a rearranged form of the lens maker's equation. Starting from 1/f = (n-1)(1/R₁ - 1/R₂), the radius of curvature calculator solves for R₂:
Radius of Curvature Calculator Formula for R₂
R₂ = 1 / [1/R₁ - 1/(f(n-1))]
Input Variables
- f = Desired focal length (meters)
- n = Material refractive index
- R₁ = First surface radius
Output
The radius of curvature calculator outputs R₂ in meters. The sign indicates surface orientation relative to the optical axis.
4. How to Use the Radius of Curvature Calculator
Follow these steps to use our radius of curvature calculator for determining R₂:
Enter desired focal length (f)
Input your target focal length in meters. The radius of curvature calculator needs this to determine the required optical power distribution.
Enter refractive index (n)
Input the refractive index of your lens material. The radius of curvature calculator uses this to account for light bending at both surfaces.
Enter first radius (R₁)
Input the first surface radius you've already determined or chosen. The radius of curvature calculator computes R₂ to complete your design.
Calculate R₂
Click calculate and the radius of curvature calculator will display the required second surface radius to achieve your target focal length.
5. Sign Convention for R₂
The radius of curvature calculator uses the Cartesian sign convention for R₂. Understanding this helps interpret radius of curvature calculator results:
R₂ < 0 (Negative)
The radius of curvature calculator gives negative R₂ when the center of curvature is to the left of the second surface (convex back surface for biconvex lens).
Common for: Biconvex, plano-convex lenses
R₂ > 0 (Positive)
The radius of curvature calculator gives positive R₂ when the center of curvature is to the right of the second surface.
Common for: Biconcave, plano-concave lenses
💡 Pro Tip
When the radius of curvature calculator gives R₂ = ∞, your design requires a flat back surface (plano lens). This is common for simple magnifying glasses.
6. Radius of Curvature Calculator Examples
Example 1: Completing a Biconvex Lens Design
Input to radius of curvature calculator:
- • Target focal length f = 0.1 m
- • Material n = 1.5 (crown glass)
- • First surface R₁ = 0.15 m (convex)
Radius of curvature calculator computation:
1/(f(n-1)) = 1/(0.1 × 0.5) = 20
1/R₁ = 1/0.15 = 6.67
R₂ = 1/(6.67 - 20) = 1/(-13.33)
Radius of curvature calculator result: R₂ = -0.075 m
Example 2: Meniscus Lens Design
Input to radius of curvature calculator:
- • Target focal length f = 0.2 m
- • Material n = 1.6
- • First surface R₁ = 0.1 m
Radius of curvature calculator computation:
1/(f(n-1)) = 1/(0.2 × 0.6) = 8.33
1/R₁ = 1/0.1 = 10
R₂ = 1/(10 - 8.33) = 1/1.67
Radius of curvature calculator result: R₂ = 0.6 m (positive meniscus)
7. Applications of the Radius of Curvature Calculator
The radius of curvature calculator for R₂ is essential in various optical design scenarios. Here's where professionals use this radius of curvature calculator:
Camera Lens Design
Use the radius of curvature calculator to design multi-element camera lenses with specific focal lengths and aberration control.
Eyeglass Lenses
The radius of curvature calculator helps opticians design lenses with proper curvature for vision correction prescriptions.
Telescope Optics
Design eyepieces and objective lenses using the radius of curvature calculator for astronomical instruments.
Scientific Instruments
The radius of curvature calculator aids in designing specialized optics for spectrometers and other lab equipment.
8. Radius of Curvature Calculator FAQ
What is the radius of curvature calculator for R₂?
The radius of curvature calculator for R₂ computes the second surface radius of a lens. Given focal length, refractive index, and R₁, this radius of curvature calculator determines the back surface curvature needed for your lens design.
How is R₂ different from R₁ in the radius of curvature calculator?
In the radius of curvature calculator, R₁ refers to the first (front) surface and R₂ to the second (back) surface. Light enters through the R₁ surface first. The radius of curvature calculator handles both surfaces but solves for different unknowns.
Why does the radius of curvature calculator give negative R₂ for biconvex lenses?
The radius of curvature calculator uses sign convention where negative R₂ indicates the center of curvature is behind the lens. For biconvex lenses, both surfaces curve outward, resulting in opposite signs from the radius of curvature calculator.
Can the radius of curvature calculator help design asymmetric lenses?
Yes! The radius of curvature calculator is perfect for asymmetric designs. Enter your known R₁, and the radius of curvature calculator will compute the specific R₂ needed regardless of symmetry, enabling optimized lens designs.