Ball lenses are great optical components for improving signal coupling between fibers, emitters and detectors. They are also used in endoscopy, bar code scanning, ball pre-forms for aspheric lenses and sensor applications. Ball lenses are manufactured from a single substrate of glass and can focus or collimate light, depending upon the geometry of the input source. Half-ball lenses are also common and can be interchanged with (full) ball lenses if the physical constraints of an application require a more compact design.

## Essential Equations for Using Ball Lenses

**Figure 1:** Key Parameters

There are five key parameters needed to understand and use ball lenses (Figure 1): Diameter of Input Source (d), Diameter of Ball Lens (D), Effective Focal Length of Ball Lens (EFL), Back Focal Length of Ball Lens (BFL) and Index of Refraction of Ball Lens (n).

EFL is very simple to calculate (Equation 1) since there are only two variables involved: Diameter of Ball Lens (D) and Index of Refraction (n). EFL is measured from the center of the ball lens, indicated by R in Figure 1. BFL (Equation 2) is easily calculated once EFL and D are known. Numerical Aperture NA (Equation 3) is dependent on EFL and d. It is a commonly referenced term and often used in lieu of d/D.

**(1)**

**(2)**

**(3)**

Since NA is often used, Figure 2 illustrates how it increases as the Diameter of the Input Source (d) also increases.

**Figure 2:** Numerical Aperture vs. Diameter for Ball Lens Glass Types offered by Edmund Optics®. [View Larger Image]

## Application Examples

**Figure 3:** Laser to Fiber Coupling

### Example 1: Laser to Fiber Coupling

When coupling light from a laser into a fiber optic, the choice of ball lens is dependent on the NA of the fiber and the diameter of the laser beam, or the input source. The diameter of the laser beam is used to determine the NA of the ball lens. The NA of the ball lens must be less than or equal to the NA of the fiber optic in order to couple all of the light. The ball lens in contact with the fiber as shown in Figure 3.

**Initial Parameters**

Diameter of Input Laser Beam = 2mm

Index of Refraction of Ball Lens = 1.517

Numerical Aperture of Fiber Optic = 0.22

**Calculated Parameter**

Diameter of Ball Lens

**(4)**

A N-BK7 ball lens (index of refraction of 1.517) of 6-8mm in diameter would be ideal for coupling a 2mm laser source into a 0.22NA fiber optic. One can easily try different indices of refraction in order to find the best ball lens for a laser to fiber coupling application.

**Figure 4:** Fiber to Fiber Coupling

### Example 2: Fiber to Fiber Coupling

To couple light from one fiber optic to another fiber optic of similar NA, two identical ball lenses are used. Place the two ball lenses in contact with the fibers as shown in Figure 4. If the fiber optics have the same NA, then the same logic as in Example 1 can be applied.