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 < ρ₂ ). 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 ≤|θ_max | 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 (θ_max ) 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 ±θ_max with respect to the fiber's axis. 

Graded-Index Multimode Fibers
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 (±θ_max ) occurs on the fiber's axis (ρ = 0). The angular spread decreases as the radial distance to the axis increases. 

Step-Index or Graded Index?
A step-index multimode fiber has the potential to collect more light than a graded-index multimode fiber. This is due to the NA being constant across the step-index core, while the NA decreases with radial distance across the graded-index core.

However, the graded-index profile causes all of the guided modes to have similar propagation velocities, which reduces the modal dispersion of the light beam as it travels in the fiber.

For applications that rely on coupling as much light as possible into the multimode fiber and are less sensitive to modal dispersion, a step-index multimode fiber may be the better choice. If the reverse is true, a graded-index multimode fiber should be considered.

References
[1] Gerd Keiser, Optical Fiber Communications (McGraw-Hill, New York, 1991), Section 2.6.