### Optipedia • SPIE Press books opened for your reference.

# Gradient Index Lens

Excerpt from Optical Design Fundamentals for Infrared Systems, Second Edition

Instead of a ball lens, a gradient index lens is frequently used as a fiber coupling element. Such a lens has a gradient profile in which the refractive index varies in the direction perpendicular to the optical axis, as expressed in Eq. (5.23).

where *N*_{0} is the base index (at the center of the lens), *k* is called the *Gradient Constant*, and *r* is the variable radius (mm). Figure 1 shows a gradient index rod (GRIN rod) with the length of one full sinusoidal path, or one “pitch.”

**Figure 1 ***A GRIN rod, one “pitch” long.*

Shortening the rod to a length of 1/4 pitch, as shown in Fig. 2, forms a lens with a focal length of

**Figure 2 ***The gradient index lens.*

Lenses of that kind are commercially available under the trade name SELFOC, a derivative from “self-focusing.”

If the length of such a lens is identified as t, then the focal length is expressed by^{1}

The back focal length is

Figure 3 shows the details of a GRIN lens with a central index of refraction of *N*_{0} = 1.5834 and a gradient constant of *k* = 0.1067. With a length *t* = 4 mm, we obtain by using Eqs. (5.25) and (5.26) a focal length of *f *= 2 mm, and a back focal length of *bfl* ≈ 0.52 mm.

**Figure 3 ***Details of a GRIN rod lens for an object located at infinity. Such an element is also known as a Wood lens.*

Figure 4 shows the same lens as Fig. 3, but now the object is located at a finite distance. To better understand the function of a GRIN lens, an “equivalent” thin lens has been superimposed for reference. This clarifies the object and image distances and, with the fields added, the magnification *m*=*s'*/*s*=*h'*/*h*.

**Figure 4 ***A GRIN lens imaging an object located at a finite distance**.*

### Reference

**Citation:**

M. Riedl, *Optical Design Fundamentals for Infrared Systems, Second Edition*, SPIE Press, Bellingham, WA (2001).

View SPIE terms of use.

Non-Member: $58.00