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Gradient Index Lens

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 N₀ 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 by¹

The back focal length is

Figure 3 shows the details of a GRIN lens with a central index of refraction of N₀ = 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

1. W. J. Smith, Modern Optical Engineering, McGraw-Hill, Inc. (2000), page 286.