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Gradient-index Lenses

Acronym: GRIN lens

Definition: lenses which utilize a radial variation of refractive index

More general term: lenses

German: Gradientenindexlinsen

Category: general optics

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The essential function of a lens is to produce a radially varying delay of the optical phase of a beam; the resulting wavefront curvature can make a beam converging or diverging after the lens. In ordinary lenses, the radially varying phase delay is produced by varying the thickness of the lens material. An alternative operation principle is that of a gradient index lens (GRIN lens), where the thickness is usually constant, while the refractive index varies in the radial direction. It is also possible (but not common) to combine both operation principles, i.e., to make GRIN lenses with curved surfaces.

Typical GRIN lenses have a cylindrical rod shape, although a wide range of other shapes is possible.

As an example, Figure 1 shows the propagation of rays (according to geometrical optics) which get deflected in a GRIN lens and may exactly meet in a focal point if the lens is optimized.

ray path at GRIN lens
Figure 1: Ray paths in and after a gradient-index lens. Within the lens (the gray area), the rays are curved.

Figure 2 shows how the beam radius, calculated with wave optics, evolves.

GRIN lens
Figure 2: Evolution of beam radius for an originally collimated Gaussian beam going through a GRIN lens. Refocusing action is not concentrated to the surfaces, but is distributed in the material.

Calculation of Dioptric Power

At least in cases where the paraxial approximation is valid, it is simple to calculate the dioptric power and focal length of a not too long gradient-index lens from its refractive index profile. The calculation is based on the fact that the radially varying phase delay caused by a lens with focal length f is given by the following equation:

radial phase change at lens

One simply needs to equate the second-order coefficients of the index profile to obtain the focal length and its inverse, the dioptric power.

Focal Length

For a gradient-index lens, being an extended optical element, it is less obvious than for a thin lens how the focal length should be defined. This requires deeper considerations, as explained in the article on focal length, see the section “Focal Length of an Extended Optical System”.

Fabrication of Gradient-index Lenses

There is a range of quite different optical fabrication methods for GRIN lenses; some examples:

  • Ion exchange methods: If a glass material is immersed into a liquid, some ions of the glass may be exchanged with other ions in the liquid, such that the refractive index is modified. Applying such a technique to the mantle of a cylindrical glass part can lead to the required refractive index profile.
  • Partial polymerization: A polymer material may be exposed to radially varying doses of ultraviolet light which causes polymerization.
  • Direct laser writing: The refractive index of various transparent media can also be changed with point-by-point laser writing, where the exposure dose is varied in the radial direction.
  • Chemical vapor deposition: Glass materials can be deposited from a chemical vapor, where the chemical composition is varied during the process such that the required index gradient is obtained.
  • Neutron irradiation can be used to generate spatially varying refractive index modifications in certain boron-rich glasses.

Applications of GRIN Lenses

GRIN lenses can be used for a wide range of applications – for example:

  • fiber collimators, where a GRIN lens may be fused to a fiber end
  • fiber-to-fiber coupling
  • mode field adapters
  • focusing applications, e.g. optical data storage
  • monolithic solid-state lasers
  • ophthalmology, e.g. for contact lenses with high dioptric power

Typical advantages of GRIN lenses are that they can be very small and that their flat surfaces allow simple mounting together with other optical components. In some cases, flat surfaces are cemented together in order to obtain a rugged monolithic setup.

If the used fabrication method allows for precise control of the radial index variation, the performance of a GRIN lens may be high, with only weak spherical aberrations similar to those of aspheric lenses.

Besides, some fabrication techniques allow for cheap mass production.

Other Devices with an Index Gradient

A radial gradient of the refractive index also often occurs in a laser crystal or other laser gain medium as a result of thermal effects. This phenomenon is called thermal lensing.

There are graded-index fibers, which in contrast to step-index fibers have a smooth variation of refractive index in the radial direction.

Questions and Comments from Users


Where is the principal plane of a GRIN lens located?

Answer from the author:

The principal plane is usually in the middle, if the index profile has no z dependence. Otherwise, you can have two different principal planes, to be calculated from the design details.

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See also: lenses, microlenses, focal length, thermal lensing, graded-index fibers
and other articles in the category general optics


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