Ball Lens Calculator
Calculate the effective focal length, back focal length, and numerical aperture of ball lenses using the formula f = nR/2(n-1). Ideal for fiber coupling, collimation, and short focal length optics.
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Effective Focal Length (EFL)
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Back Focal Length (BFL)
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Numerical Aperture (NA)
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Ball Lens Formula
Effective Focal Length
f = nR / 2(n − 1)
Back Focal Length
BFL = R(2 − n) / 2(n − 1)
f — effective focal length (mm or m, same units as R)
n — refractive index
R — radius of the ball lens
BFL — distance from rear surface to focal point
Derivation from Lens Maker Equation
A ball lens is a thick spherical lens: both surfaces have radius R, with R₁ = +R (light enters convex) and R₂ = −R (light exits concave). The thickness is d = 2R.
The thick lens maker equation is:
Substituting R₁ = R, R₂ = −R, d = 2R:
1/f = (n−1)[1/R − 1/(−R) + (n−1)·2R/(n·R·(−R))]
= (n−1)[2/R − 2(n−1)/(nR)]
= (n−1)·2[1/R − (n−1)/(nR)]
= 2(n−1)·[n − (n−1)]/(nR)
= 2(n−1)/(nR) → f = nR/2(n−1)
The back focal length BFL = f − R follows from the geometry of the spherical lens. Simplifying: BFL = R(2−n)/2(n−1).
Lens Maker Equation →Applications
Fiber coupling
Couple light between optical fibers with minimal loss. Ball lenses provide easy alignment and high NA.
Collimation
Collimate divergent light from lasers or LEDs into a parallel beam for illumination or sensing.
Endoscopy
Miniature ball lenses are used in medical endoscopes and industrial borescopes for imaging.
Sensors & scanning
Barcode scanners, optical sensors, and confocal microscopy use ball lenses for compact optics.
Common Ball Lens Materials
| Material | n | Transmission Range |
|---|---|---|
| BK7 | 1.517 | 350 nm – 2 μm |
| Sapphire | 1.768 | 150 nm – 5.5 μm |
| Fused Silica | 1.458 | 180 nm – 2.2 μm |
| LaSFN9 | 1.85 | 400 nm – 2.5 μm |
| N-SF11 | 1.785 | 365 nm – 2.5 μm |
| Ruby | 1.77 | 400 nm – 3 μm |
| Zinc Selenide | 2.4 | 600 nm – 18 μm |
Frequently Asked Questions
What is the focal length formula for a ball lens?
The effective focal length of a ball lens is f = nR/2(n-1), where n is the refractive index and R is the radius of the ball. The back focal length (distance from the rear surface) is BFL = f - R = R(2-n)/2(n-1).
How is the ball lens formula derived from the lens maker equation?
A ball lens is a thick lens with R₁ = R and R₂ = -R (sphere). Applying the thick lens maker equation with thickness d = 2R gives 1/f = 2(n-1)/(nR), so f = nR/2(n-1).
What are ball lenses used for?
Ball lenses are used for fiber-to-fiber coupling, laser collimation, endoscopy, barcode scanning, and sensor applications. Their symmetric shape makes alignment easy and they work well for short focal length applications.
What is numerical aperture (NA) for a ball lens?
NA ≈ R/f is a useful approximation for ball lenses, relating the radius and focal length. Higher NA means greater light collection and coupling efficiency.
When is BFL negative?
BFL = R(2-n)/2(n-1) is negative when n > 2. For most optical glasses (n ≈ 1.45–1.85), BFL is positive and the focal point lies outside the sphere.