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Author: the photonics expert

Acronym: GRIN lens

Definition: lenses which utilize a radial variation of refractive index

More general term: lenses

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.

## Optical Properties

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.

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

Note that there are also defocusing GRIN lenses, where the refractive index increases in the radial direction; they can be used for micro-optical telescopes, for example. Besides, one can produce cylindrical GRIN lenses.

### 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:

$$\Delta \varphi (r) = - \frac{\pi }{{\lambda f}}{r^2}$$

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”.

### Pitch of a GRIN Lens

A parallel input ray leads to an oscillatory (approximately sinusoidal) propagation path in a focusing GRIN lens. The pitch of a focusing GRIN lens is defined as the number of oscillation cycles of such a ray which can occur over the whole length. For example, a half-pitch lens is one where the output ray occurs just on the opposite side of the center at the same radial position. It actually images an object on the entrance surface to the exit surface with an inversion. A full-pitch lens does such imaging without inversion. A quarter-pitch lens can be used as a beam collimator, for example. Of course, not only specific versions like quarter-pitch, half-pitch etc. are used for various applications, but basically any pitch values from very small ones to 1 or even larger. With a very small pitch value, one approaches the limiting case of a thin lens.

### Numerical Aperture

The numerical aperture of a lens is related to the maximum acceptance angle. For a GRIN lens, that is determined by the maximum refractive index change of the transverse profile. The achievable index change without excessive side effects such as propagation losses by absorption depends on the used fabrication technology (see below). For example, lithium ion exchange is limited to relatively low NA values (≈0.2), while silver ion exchange allows for substantially higher values around 0.5.

### Optical Aberrations

GRIN lenses, similar to other types of lenses, exhibit some amount of optical aberrations such as spherical aberrations and chromatic dispersion. There magnitude can strongly depend on the used fabrication method.

### Parasitic Reflections

As for other types of lenses, parasitic reflections can occur on the end faces of GRIN lenses (→ Fresnel reflection). One generally uses anti-reflection coatings in order to suppress such reflections as far as possible.

### Birefringence

Unfortunately, the refractive index profile can also be accompanied by some level of birefringence [12], e.g. induced by stress in lenses fabricated with the ion exchange method. This can have detrimental effects e.g. in imaging applications. The amount of birefringence depends on the fabrication method, and such methods are sometimes optimized to minimize birefringence.

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. Thallium technology has been widely used, but thallium is more and more replaced with non-toxic agents such as lithium or silver ions. Note that such dopants generally increase absorption in certain spectral regions, which may lead to a color appearance, but with a suitable choice of glass such effects can be minimized.
• Partial polymerization: A polymer material (→ plastic optics) 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.

Different methods are suitable for fabricating gradient-index lenses with different diameters, which are typically be between a few hundred microns and several millimeters.

## 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
• focusing applications, e.g. optical data storage
• monolithic solid-state lasers
• ophthalmology, e.g. for contact lenses with high dioptric power
• imaging applications, e.g. objectives for endoscopes

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.

## More to Learn

Encyclopedia articles:

## Suppliers

The RP Photonics Buyer's Guide contains nine suppliers for gradient-index lenses. Among them:

## Questions and Comments from Users

2020-07-09

Where is the principal plane of a GRIN lens located?

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.