Ball lens

A ball lens is a sphere of glass, sapphire or another transparent material used as a very short focal length lens. Both surfaces share one center of curvature, which makes the lens symmetric in every direction: it has no optical axis of its own, so it cannot be tilted and needs to be positioned only in three translations. Ball lenses with diameters of about 0.5–10 mm are made in large volumes to tight diameter tolerances and are used to couple light between fibers, from laser diodes into fibers, and into photodiodes, often mounted directly in the cap of a transistor-outline package. Half-balls and drum lenses, spheres ground with flat faces, are variants that are easier to hold.

Focal length formulas

For a sphere of diameter $D$ and index $n$ in air, the effective focal length, measured from the center of the sphere, and the back focal length, measured from its surface, are

$$\mathrm{EFL} = \frac{n D}{4(n-1)} ,$$ $$\mathrm{BFL} = \mathrm{EFL} - \frac{D}{2} .$$

For a 2 mm N-BK7 ball ($n_d = 1.5168$), EFL = 1.47 mm and BFL = 0.47 mm. A 2 mm sapphire ball ($n \approx 1.768$) has EFL = 1.15 mm and BFL = 0.15 mm. As the index approaches 2 the paraxial focus moves onto the back surface (EFL = $D/2$, BFL = 0), which is why high-index glasses near $n = 2$ are chosen when a fiber end is to sit in contact with the ball. Because the focal length scales with $D$, a 1 mm N-BK7 ball has EFL = 0.73 mm and a 5 mm ball has BFL = 1.17 mm.

Spherical aberration

A sphere is a strongly aberrated lens: rays striking it far from the center line are refracted too much and cross the axis short of the paraxial focus. Exact ray tracing through the 2 mm N-BK7 ball shows the size of the spherical aberration:

The shift is measured toward the ball from the paraxial focus. For comparison, the diffraction depth of focus $\lambda/\mathrm{NA}^2$ at 632.8 nm is about 22 µm for a 0.5 mm diameter beam (NA about 0.17) and about 5 µm for a 1 mm beam (NA about 0.34). The ball is therefore close to diffraction-limited only when the beam fills about a quarter of its diameter; with a larger fill the focus is dominated by aberration. This is the main trade-off against an aspheric lens, which reaches higher usable numerical aperture without the aberration penalty but costs more and must be aligned in tilt.

Coupling applications

• Fiber to fiber. Two identical ball lenses, each with a fiber at its focus, form an expanded-beam connection: the beam between them is several hundred micrometres wide and much less sensitive to dust and lateral misalignment than a direct butt joint. Expanded-beam connectors for harsh environments use this principle.
• Laser diode to fiber. A single ball can image the diode's emitting facet onto a fiber core at a magnification chosen by the object and image distances, trading some efficiency for low cost and simple assembly.
• Detectors and LEDs. Ball lenses in package caps concentrate light onto small photodiodes and collect light from LEDs.
• Endoscopes and sensors. Sapphire balls serve as robust window-lenses at the tips of probes, combining the functions of seal and lens.

Measurement and alignment

The diameter is specified to a few micrometres or better, and since the focal length follows directly from the diameter and index, catalog focal lengths are computed rather than measured. In use, the fiber or source is scanned in three axes while monitoring coupled power; for single-mode fiber, a lateral offset of about half the mode field radius, some 2.6 µm for a 10.4 µm mode field diameter, already costs 1 dB when the focused spot matches the fiber mode. Ball lenses for laser use carry anti-reflection coatings applied over the whole sphere, since any part may face the beam.

Common questions

How is the focal length of a ball lens calculated?

Use EFL = $nD/(4(n-1))$, measured from the center, and subtract the radius to get the back focal length from the surface. A 2 mm N-BK7 sphere gives 1.47 mm and 0.47 mm. The formula is paraxial; marginal rays focus closer, as the table above shows.

Why is the focal length of a ball lens so short?

Both surfaces are strongly curved, with a radius equal to half the diameter, and both bend the light the same way. For ordinary glass this puts the focus less than half a diameter behind the sphere.

When is a GRIN lens better than a ball lens?

A GRIN lens has flat faces that can be bonded to a fiber and angle-polished to control back reflection, and it has lower aberration for fiber collimators that need a clean, well-collimated beam. A ball lens is cheaper, needs no rotational alignment, and suits short-distance coupling where some aberration loss is acceptable.

References: E. Hecht, Optics, 5th ed. (Pearson, 2017), Ch. 5 and 6; W. J. Smith, Modern Optical Engineering, 4th ed. (McGraw-Hill, 2008); B. E. A. Saleh, M. C. Teich, Fundamentals of Photonics, 3rd ed. (Wiley, 2019), Ch. 1.