Sign Convention (Cartesian):
Light travels left to right.
• Convex front surface: R1 > 0
• Concave front surface: R1 < 0
• Concave back surface: R2 > 0
• Convex back surface: R2 < 0 (Standard bi-convex lens: R1 +, R2 -)
Front surface curvature
Back surface curvature
Axial thickness of lens
Material index (Air ≈ 1.0)
Effective Focal Length (EFL):–
Optical Power:–
Back Focal Length (BFL):–
Front Focal Length (FFL):–
System Focal Points:–
Understanding Thick Lenses
In optics, the distinction between a "thin lens" and a "thick lens" is crucial for precision engineering and high-quality imaging systems. While a thin lens approximation assumes the thickness of the lens is negligible compared to the radii of curvature, a thick lens model accounts for the physical distance light travels through the refractive material.
This calculator utilizes the Gullstrand equation (Lensmaker's Equation for thick lenses) to determine the effective focal length (EFL), optical power, and back focal length (BFL). These parameters are essential when designing eyepieces, camera objectives, or corrective lenses where the glass thickness significantly impacts the path of light.
The Physics and Formulas
The calculations performed above are based on Gaussian optics. The total optical power ($\Phi$) of a thick lens in air is derived from the individual powers of the front and back surfaces, adjusted for the thickness separating them.
Key Definitions:
Radius R1: The radius of curvature of the front surface (incident side).
Radius R2: The radius of curvature of the back surface (emergent side).
Thickness (t): The distance between the vertices of the two surfaces along the optical axis.
Refractive Index (n): A property of the lens material (e.g., BK7 glass is approx 1.517) indicating how much it bends light.
Where $\Phi_1 = \frac{n-1}{R_1}$ and $\Phi_2 = \frac{1-n}{R_2}$.
Once the power is known, the Effective Focal Length (EFL) is simply $f = 1/\Phi$. The Back Focal Length (BFL), which is the physical distance from the back surface of the glass to the focal point, is calculated by adjusting the EFL based on the position of the second principal plane.
Common Applications
Thick lens calculations are vital in:
Eyewear: High-prescription lenses are physically thick, requiring compensation to ensure the focal point hits the retina correctly.
Microscopy: Objective lenses are complex thick lens systems composed of multiple elements.
Lasers: Focusing collimated beams requires precise knowledge of BFL to prevent damage to optical components.
Sign Conventions
This calculator uses the standard Cartesian sign convention. Light is assumed to travel from left to right. A surface convex to the incoming light has a positive radius (center of curvature is to the right). A surface concave to the incoming light has a negative radius. Consequently, a standard biconvex lens (bulging out on both sides) will have a positive R1 and a negative R2.
function calculateThickLens() {
// 1. Get Inputs
var r1 = parseFloat(document.getElementById('radius1').value);
var r2 = parseFloat(document.getElementById('radius2').value);
var t = parseFloat(document.getElementById('thickness').value);
var n = parseFloat(document.getElementById('refractiveIndex').value);
// 2. Validation
if (isNaN(r1) || isNaN(r2) || isNaN(t) || isNaN(n)) {
alert("Please enter valid numerical values for all fields.");
return;
}
if (n <= 1) {
alert("Refractive index must be greater than 1.");
return;
}
if (t < 0) {
alert("Thickness cannot be negative.");
return;
}
// Handle flat surfaces (Infinite Radius)
// If user enters 0, we can't divide by zero.
// In optics, Power = (n-1)/R. If R is infinity, Power is 0.
// We calculate surface powers first to handle this safely.
var power1, power2;
// Surface 1 Power (Front)
if (Math.abs(r1) < 0.0001) {
// Treating 0 as planar is tricky in code, usually users put large num or 0 means error.
// We will assume 0 means "Infinite/Flat" if specific checkbox checked, but here we assume user inputs numbers.
// Let's assume input is non-zero for curvature. If they want flat, enter 1e9.
// If they actually enter 0, it crashes math. Let's guard.
alert("Radius cannot be exactly zero. For a flat surface, enter a very large number (e.g. 999999).");
return;
} else {
power1 = (n – 1) / r1; // Units: 1/mm
}
// Surface 2 Power (Back)
// Note: Formula for surface 2 usually (1-n)/R2 if using same sign convention
// Ref: Hecht Optics. Power = (n2-n1)/R. Surface 2 goes from n to air (1).
// So (1 – n) / R2.
if (Math.abs(r2) < 0.0001) {
alert("Radius cannot be exactly zero. For a flat surface, enter a very large number (e.g. 999999).");
return;
} else {
power2 = (1 – n) / r2; // Units: 1/mm
}
// 3. Calculate System Power (Gullstrand Equation)
// P = P1 + P2 – (t/n)*P1*P2
var term3 = (t / n) * power1 * power2;
var totalPower = power1 + power2 – term3; // Units: 1/mm
// 4. Calculate Focal Lengths
// Effective Focal Length (EFL) f
var efl = 0;
if (Math.abs(totalPower) 0) typeText = "Converging (Positive)";
else if (efl < 0) typeText = "Diverging (Negative)";
else typeText = "Afocal (Zero Power)";
document.getElementById('res-type').innerHTML = typeText;
document.getElementById('results').style.display = 'block';
}