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Thin Lenses ... Converging Lens. Parallel light incident on one side of a converging lens passes through a single point on the other side called the "focal point" ...

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.”

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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.

GRINlenspitch

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GRINlensInscopix

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.

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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).

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