GRIN lens

A gradient-index (GRIN) lens is a cylindrical rod, usually of glass, with flat, parallel end faces and a refractive index that is highest on the axis and decreases toward the edge. Rays in the rod bend continuously toward the high-index axis and follow sinusoidal paths, so the rod focuses light through its internal index profile rather than through curved surfaces. Rods of about 0.5–3 mm diameter and a few millimetres long, with numerical apertures of roughly 0.1–0.6, are standard parts in fiber-optic components, where the flat faces can be polished, coated and bonded directly to fibers and filters.

Index profile and pitch

The profile is close to parabolic:

$$n(r) = n_0\left(1 - \tfrac{1}{2} g^2 r^2\right),$$

where $n_0$ is the axial index and $g$ (also written $\sqrt{A}$) is the gradient constant in mm⁻¹. In the paraxial limit a ray entering at height $r_0$ with internal slope $r_0'$ follows

$$r(z) = r_0 \cos gz + \frac{r_0'}{g}\sin gz ,$$

a sinusoid with period, called the pitch,

$$P = \frac{2\pi}{g} .$$

The length of a rod is quoted as a fraction of $P$. A quarter-pitch rod turns a point source on its front face into a collimated beam at the back face, and a collimated beam into a focus on the face; a half-pitch rod images the front face inverted onto the back face at unit magnification; a full pitch gives an erect image. Rods slightly shorter than a quarter pitch, such as 0.23 pitch, put the focus a short working distance outside the face, which leaves room for a fiber ferrule or an air gap.

Worked example

For illustrative parameters of the order of those in catalog rods, $n_0 = 1.60$, $g = 0.33$ mm⁻¹ and diameter $2a = 1.8$ mm:

• pitch $P = 2\pi/g = 19.0$ mm, so a quarter-pitch rod is 4.76 mm long;
• effective focal length of a quarter-pitch rod, $f = 1/(n_0 g) = 1.89$ mm;
• numerical aperture, from the largest ray that stays within the rod, $\mathrm{NA} \approx n_0 g a = 0.48$.

A single-mode fiber at the focus of this rod, with a mode field diameter of about 10 µm, produces a collimated beam roughly $2 f \times \mathrm{NA}_\text{mode}$ across, a few hundred micrometres for the typical fiber mode NA of about 0.1. For a rod of any length $L$ the focal length is $f = 1/(n_0 g \sin gL)$, which reduces to the value above when $gL = \pi/2$.

How the gradient is made and measured

Glass GRIN rods are made by ion exchange: a base-glass rod is held in a molten salt bath so that ions diffuse in and out through the surface, and the resulting composition profile sets the index profile. Profiles are measured by refracted near-field or interferometric methods, and the pitch is checked optically at the working wavelength by locating the focus of a collimated beam.

Where GRIN lenses are used

• Fiber collimators. A quarter-pitch rod bonded to a fiber ferrule forms a compact fiber collimator, and pairs of them bracket free-space elements in fiber isolators, circulators, filters and switches; the optical isolator in a fiber-pigtailed package is usually built this way.
• Endoscopes and probes. Long multi-pitch rods relay images through thin rigid endoscopes, and GRIN objectives a millimetre across are used in miniature microscopes and fiber probes.

Pitfalls

The gradient constant depends on wavelength, so the pitch, working distance and focal length are specified at one wavelength and drift at others; a rod cut to quarter pitch at 1550 nm is not at quarter pitch at 1310 nm. Real profiles depart from the ideal, which introduces aberrations that grow toward the edge of the rod, so imaging quality is best near the axis and at moderate NA. The flat faces reflect like any glass surface and would send light straight back into the fiber, so fiber-coupled rods carry anti-reflection coatings and are often polished at an angle of about 8°, which in turn tilts and offsets the output beam slightly. GRIN rods are also hard to obtain in long focal lengths and large apertures, where a conventional lens is used instead.

Common questions

What does the pitch of a GRIN lens mean?

The pitch is the length over which a ray completes one full sinusoidal period in the rod, $P = 2\pi/g$. Rod lengths are given as fractions of it: 0.25 pitch for collimation or focusing, 0.5 pitch for relaying an inverted image, 1 pitch for an erect image.

Why use a GRIN lens instead of a ball lens?

The flat faces can be polished at an angle, coated and glued directly to a fiber ferrule or a filter, and the rod fits a cylindrical sleeve without separate mounting. A ball lens is cheaper and has a higher NA for its size, but its strong spherical aberration and point contact make it harder to package for collimation.

Is a GRIN lens the same as graded-index fiber?

They use the same physics, a radial index gradient that bends rays back toward the axis. A GRIN lens has a strong gradient and a length of a fraction of a pitch; a graded-index fiber has a much weaker gradient and runs for hundreds of metres or more, where the aim is to equalize the travel time of its modes.

References: B. E. A. Saleh, M. C. Teich, Fundamentals of Photonics, 3rd ed. (Wiley, 2019), Ch. 1; E. Hecht, Optics, 5th ed. (Pearson, 2017), Ch. 6; E. W. Marchand, Gradient Index Optics (Academic Press, 1978).