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:

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

where $\Delta = (n_1^2 - n_2^2)/(2n_1^2)$ is the relative index difference and $\alpha$ is the profile parameter. For $\alpha = 2$ the profile is parabolic; for $\alpha \to \infty$ 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 ($v = c/n$), 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:

$$\Delta \tau / L \;\approx\; \frac{\Delta^2}{8c/n_1}.$$

For $\Delta = 0.01$ ($n_1 = 1.50$, $n_2 = 1.485$): residual modal dispersion $\sim 0.06$ ns/km — about 100× smaller than step-index multimode fiber.

Optimum profile parameter. The exact $\alpha$ that minimizes modal dispersion is wavelength-dependent and slightly less than 2:

$$\alpha_\text{opt} \;\approx\; 2 \left(1 - \frac{2 \lambda \, n_1}{N_1 \, \Delta} \frac{dN_1}{d\lambda}\right),$$

where $N_1$ is the group index. For silica-based fibers at 850 nm: $\alpha_\text{opt} \approx 2.03$. At 1300 nm: $\alpha_\text{opt} \approx 2.01$. Standard OM3/OM4/OM5 multimode fiber is manufactured with $\alpha \approx 2.0$, optimized as a compromise across the operating wavelength range.

Standard graded-index multimode fibers.

Standard	Core diameter	NA	Bandwidth at 850 nm	Bandwidth at 1300 nm
OM1 (62.5/125, legacy)	62.5 μm	0.275	200 MHz·km	500 MHz·km
OM2 (50/125)	50 μm	0.20	500 MHz·km	500 MHz·km
OM3 (50/125, laser-optimized)	50 μm	0.20	2000 MHz·km	500 MHz·km
OM4 (50/125, enhanced)	50 μm	0.20	4700 MHz·km	500 MHz·km
OM5 (50/125, wide-band)	50 μm	0.20	3500 MHz·km (850 nm) + 1850 MHz·km (953 nm)	—

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 $\Delta \alpha \pm 0.005$ 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's no advantage to graded-index in single-mode systems, so step-index is universally used.

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 requires expensive cooled DFBs or external modulators), multimode is dramatically cheaper for short distances. Single-mode dominates for $> 100$ m.

References: 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.