# Numerical Aperture of GRIN Multimode Fibers Varies Across the Core

## Is the max acceptance angle constant across the core of a multimode fiber?

It depends on the type of fiber. A step-index multimode fiber provides the same maximum acceptance angle at every position across the fiber's core. Graded-index multimode fibers, in contrast, accept rays with the largest range of incident angles only at the core's center. The maximum acceptance angle decreases with distance from the center and approaches 0° near the interface with the cladding.

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Figure 1: Step-index multimode fibers have an index of refraction ( n ) that is constant across the core. Graded-index multimode fibers have an index that varies across the core. Typically the maximum index occurs at the center.

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Figure 3: Graded-index multimode fibers have acceptance angles that vary with radius ( ρ ), since the refractive index of the core varies with radius. The largest acceptance angles typically occur near the center, and the smallest, which approach 0°, occur near the boundary with the cladding (0 < ρ1 < ρ). Air is assumed to surround the fiber.

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Figure 2: Step-index multimode fibers accept light incident in the core at angles ≤|θmax | with good coupling efficiency. The maximum acceptance angle is constant across the core's radius ( ρ ). Air is assumed to surround the fiber.

Step-Index Multimode Fiber
The core of a step-index multimode fiber has a flat-top index profile, which is illustrated on the left side of Figure 1. When light is coupled into the planar end face of the fiber, the maximum acceptance angle (θmax ) is the same at every location across the core (Figure 2). This is due to the constant value of refractive index across the core, since the acceptance angle depends strongly on the index of the cladding.

Regardless of whether rays are incident near the center or edge of the core, step-index multimode fibers will accept cones of rays spanning angles ±θmax with respect to the fiber's axis.

The core of a typical graded-index multimode fiber, shown on the right side of Figure 1, has a refractive index that is greatest at the center of the core and decreases with radial distance ( ρ ). The equation included below the diagram in Figure 3 shows that the radial dependence of the core's refractive index results in a radial dependence of the maximum acceptance angle and numerical aperture (NA). This equation also assumes a planar end face, normal to the fiber's axis that is surrounded by air.

Cones of rays with angular ranges limited by the core's refractive index profile are illustrated Figure 3. The cone of rays with the largest angular spread θmax ) occurs on the fiber's axis (ρ = 0). The angular spread decreases as the radial distance to the axis increases.