Graded-index fiber

A graded-index (GI) fiber has a refractive-index profile $n(r)$ that varies continuously with radial position $r$, rather than the abrupt step of step-index fiber. The standard profile is a power-law:

$$n(r)=\{\begin{array}{ll}{n}_1\sqrt{1-2\mathrm{\Delta }(r\mathrm{/}a{)}^{\alpha }}& r<a\\ n_2=n_1\sqrt{1-2\mathrm{\Delta }}& r>a\end{array}$$

where $\mathrm{\Delta }=(n_1^2-n_2^2)\mathrm{/}(2n_1^2)$ is the relative index difference and $\alpha$ is the profile parameter. For $\alpha =2$ the profile is parabolic; for $\alpha \to \mathrm{\infty }$ it degenerates to step-index.

Why graded-index reduces modal dispersion. In step-index multimode fiber, the highest-order modes propagate at angles close to the critical angle, traveling longer geometric paths than the fundamental mode. This produces large modal dispersion (pulse broadening from differential mode delay).

In a graded-index fiber, light traveling at angles to the fiber axis spends more time in regions of lower refractive index (toward the cladding). Lower index means higher phase velocity ($v=c\mathrm{/}n$), which compensates for the longer geometric path. For the special case of $\alpha \approx 2$ (parabolic profile), this compensation is nearly exact: all bound modes propagate with the same group velocity to leading order.

The remaining modal dispersion in optimally-designed graded-index fiber comes from higher-order corrections; the dispersion residual is approximately:

$$\mathrm{\Delta }\tau \mathrm{/}L\approx \frac{{\mathrm{\Delta }}^2}{8c\mathrm{/}{n}_1}\mathrm{.}$$

For $\mathrm{\Delta }=0.01$ ($n_1=1.50$, $n_2=1.485$): residual modal dispersion $\sim 0.06$ ns/km, about 800× smaller than the $n_1\mathrm{\Delta }\mathrm{/}c\approx 50$ ns/km of step-index multimode fiber with the same $\mathrm{\Delta }$.

Optimum profile parameter. The profile that minimizes modal dispersion is close to, but not exactly, parabolic. Olshansky and Keck found the optimum

$$\begin{array}{rl}{\displaystyle {\alpha }_{\text{opt}}}& {\displaystyle =2+\epsilon -\mathrm{\Delta }\frac{(4+\epsilon )(3+\epsilon )}{5+2\epsilon },}\\ {\displaystyle \epsilon }& {\displaystyle =-\frac{2n_1}{N_1}\frac{\lambda }{\mathrm{\Delta }}\frac{d\mathrm{\Delta }}{d\lambda },}\end{array}$$

where $N_1$ is the group index of the core. The term $\epsilon$ is the profile dispersion: $\mathrm{\Delta }$ changes with wavelength because the dopant raises the index of the core and also changes its material dispersion relative to the silica cladding. Without profile dispersion the optimum is $2-12\mathrm{\Delta }\mathrm{/}5$, or 1.976 for $\mathrm{\Delta }=0.01$. With it, the optimum depends on the dopant and on the wavelength, so a fiber graded for peak bandwidth at one wavelength has less at another. Laser-optimized OM3 and OM4 fibers are graded for 850 nm; OM5 is graded to keep a high bandwidth from 850 to 953 nm.

Standard graded-index multimode fibers.

OM3 and OM4 are designed specifically for 850 nm VCSEL-launched applications (10G-SR, 40G-SR4, 100G-SR4 Ethernet). OM5 extends usable wavelength range to enable short-wavelength wavelength-division multiplexing (SWDM).

Refractive-index profile measurement. Standard techniques:

• Refracted near-field (RNF): probe fiber with HeNe laser; measure refracted intensity from each radial position
• Transmitted near-field with image processing: image the core under illumination with edge-detection
• Spatial spectral interferometry: white-light interferometer scan reveals profile

Modern multimode fibers achieve $\alpha =2.0\pm 0.005$ tolerance.

Manufacturing. Graded-index fibers are produced by varying the dopant concentration during preform deposition:

1. Inside-tube deposition (MCVD) or outside-tube deposition (OVD) builds up the preform with controlled GeO₂ dopant concentration
2. Concentration is varied across the core to produce the desired index profile
3. Standard SiO₂ tubes serve as preform cladding
4. Preform is collapsed and drawn into fiber

Manufacturing tolerances of $\mathrm{\Delta }\alpha \pm 0.005$ require very precise dopant concentration control. The challenge of achieving the exact optimum profile is the primary reason graded-index fiber is more expensive than step-index single-mode fiber.

Why single-mode is still step-index. Single-mode operation requires $V<2.405$; the mode shape is the LP01 Gaussian-like distribution that does not benefit from index grading. There is little advantage to index grading in single-mode fiber, so standard single-mode fiber uses a step or near-step core profile.

Why multimode persists in datacom. Graded-index multimode fiber, paired with low-cost 850 nm VCSELs, provides cost-effective 10G/40G/100G short-reach (up to 100 m) interconnect in data centers. Compared to single-mode fiber (which needs more expensive single-mode lasers and tighter alignment tolerances), multimode is dramatically cheaper for short distances. Single-mode dominates for $>100$ m.

The Ethernet reach of each multimode grade, and how the grades mix, are covered in Types of Optical Fiber.

References: R. Olshansky and D. B. Keck, "Pulse broadening in graded-index optical fibers," Applied Optics 15, 483 (1976); Saleh & Teich, Fundamentals of Photonics (3rd ed., 2019), Ch. 9; Snyder & Love, Optical Waveguide Theory (Chapman & Hall, 1983), Ch. 13 (graded-index modal analysis); TIA-492AAAD for OM4 specifications; IEC 60793-2-10 for international multimode standards.