# Acceptance Angle in Fiber Optics

Author: the photonics expert Dr. Rüdiger Paschotta

Definition: the maximum incidence angle of a light ray which can be used for injecting light into a fiber core or waveguide

The acceptance angle of an optical fiber is defined based on a purely geometrical consideration (ray optics): it is the maximum angle of a ray (against the fiber axis) hitting the fiber core which allows the incident light to be guided by the core since total internal reflection can occur at the core–cladding boundary. For larger incidence angles, there is no total internal reflection, and therefore there are significant power losses at each reflection point.

All accepted directions together form a *cone of acceptance*.

## Geometric Calculation of a Fiber's Acceptance Angle

The acceptance angle is usually calculated in the simplest possible situation, assuming a step-index fiber and an incident ray hitting the center of the fiber's input end. Further, one ignores the curvature of the core–cladding interface, applying the well-known equation equation for total internal reflection for a plane interface between two materials with different refractive indices. The medium from which the incident ray comes is not necessarily air or vacuum; one can more generally assume a homogeneous medium with some refractive index <$n_0$>.

The simple calculation leads to the following formula for the acceptance angle:

$${\theta _{{\rm{acc}}}} = \arcsin \left( {\frac{1}{n_0} \sqrt {n_{\rm{co}}^2 - {n_{\rm{cl}}}^{\rm{2}}} } \right)$$Here, <$n_\textrm{co}$> and <$n_\textrm{cl}$> are the refractive indices of core and cladding.

The term <$\sqrt {n_{\rm{co}}^2 - {n_{\rm{cl}}}^{\rm{2}}}$> is called the numerical aperture, and it is essentially determined by the refractive index contrast between core and cladding of the fiber.

For larger incidence angles, there is no total internal reflection, and much of the incident light will *not* be reflected at the core–cladding boundary. It will thus get into the cladding and will then usually experience strong propagation losses particularly at the outer part of the cladding.

## Considering Wave Optics

Geometric rays are generally a poor approximation for light beams when the dimensions gets small, as they typically do for optical waveguides. In general, one needs to consider wave optics. A real light beam (for example, a laser beam) is not well resembled by a ray, since it inevitably has both a finite beam radius and a finite beam divergence.

Nevertheless, for a strongly multimode waveguide, the acceptance angle as calculated above can be used to estimate the maximum input angle of a laser beam for which a high launch efficiency of the waveguide can be achieved. For single-mode fibers, however, this rule provides at most a very rough estimate.

In reality, there is not a well-defined transition between guidance and non-guidance, when a beam angle is varied; the launch efficiency varies gradually. Only in the limit of a highly multimode waveguide, such estimates based on geometrical optics become reasonably accurate. See the case study below.

## Case Study: The Numerical Aperture of a Fiber: a Strict Limit for the Acceptance Angle?

The requirement of total internal reflection would seem to set a strict limit for the angular distributions of fiber modes. However, some modes are found to exceed that limit significantly. We investigate that in detail for single-mode, few-mode and multimode fibers.

## Acceptance Angle in Nonlinear Optics

Note that the term *acceptance angle* also plays a role in nonlinear optics – see the article on critical phase matching. Here, that terms has a quite different meaning.

## More to Learn

Encyclopedia articles:

## Questions and Comments from Users

2021-03-13

How do you calculate the maximum acceptance angle in water and in air for the case of an uncladded fiber?

The author's answer:

If you just mean an optically homogeneous fiber, not having core and cladding, and you can regard the surrounding air or water as your cladding. Just use the calculator above.

2021-04-07

Does the angle of exit of light from a step-index multimode fiber tell you anything about the angle of entry?

The author's answer:

Theoretically, that angle should be preserved, but any bending of the fiber or imperfections of the fiber structure may easily spoil that relation.

2021-09-04

Why is the acceptance angle not dependent on the diameter? Isn't there any diffraction effects? Or doesn't the divergence angle of the Gaussian beam come into play?

The author's answer:

The acceptance angle is based on purely geometric reasoning (geometrical optics). Diffraction is not considered in that context. Within wave optics, where diffraction can be considered, there is no precisely defined acceptance angle.

2022-08-01

Does the acceptance angle depend on the wavelength of light?

The author's answer:

Only weakly through the refractive index.

2023-03-07

Is the maximum acceptance angle equal to the maximum possible angle of the output beam of the fiber?

The author's answer:

Geometrically yes, but such a light beam is of course not a realistic description of what can come out of such a fiber – particularly if it is a single-mode or few-mode fiber.

2023-03-18

Is the acceptance angle the same as the field of view of the fiber?

The author's answer:

The term field of view is a quantity defined for optical systems, often for imaging systems, and should not be applied to an optical fiber.

2023-11-28

For optimum coupling of a focused light beam in a multimode fiber (with focal spot size < fiber core diameter), what should be the relationship between the cone angle of the focused beam and acceptance angle of the fiber, i.e., by what percentage should the cone angle be less than the acceptance angle?

The author's answer:

That depends on what exactly you mean with “cone angle”. Ideally, the whole angular distribution would be within the acceptance angle according to the NA.

2024-04-21

If you have a cladding and core refractive index so large that NA > 1 so that sin(acceptance angle) does not compute, what does this mean physically for the acceptance angle? An example would be a SiO_{2} cladding with <$n$> = 1.45 and SiN_{x} core with <$n$> = 2 (at 850 nm).

The author's answer:

This means that even for the largest possible incidence angles of rays (in air!) you will get total internal reflection at the core–cladding boundary.

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2020-07-04

If a fiber is immersed in water, how does that change its acceptance angle?

The author's answer:

The acceptance angle is substantially reduced, as you can calculate with the formula given above.