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Numerical Aperture / Effective Numerical Aperture

Effective Numerical Aperture

Numerical aperture

Effective numerical aperture NAe2

For fiber-coupling purposes an effective fiber NAe2 defined at the 1/e2-level is more convenient than the nominal fiber NA defined by the refractive indices since Gaussian beams generally are defined by their 1/e2 diameter, also.
Schäfter+Kirchhoff defines an effective fiber NA which corresponds to the divergence of the power distribution emitted by the fiber taken at the 1/e2-level of the Gaussian angle distribution.
This NA value is the designated effective numerical aperture NAe2. For a typical single-mode fiber the value is NAe2 = 0.075.
For single-mode fibers and for polarization-maintaining fibers, the effective NAe2 typically decreases slightly with increasing wavelength λ.

1 Single-mode fiber
2      Core diameter
3 Mode-field diameter
4 Intensity level (13.5% or 1/e2-level)
5 Intensity level (e.g. 5%)
6 2xNA (e.g. 5%)
7 2xNAe2 (13.5% or 1/e2-level)

Typical Values for the effective numerical Aperture

Please find here some measured (averaged) values for the effective numerical aperture NAe2 for common fibers at their respective cut-off wavelength.

When purchasing a fiber from Schäfter+Kirchhoff, the fiber is delivered with more accurate measurements of the effective numerical aperture NAe2. Schäfter+Kirchhoff determines the NAe2 of the fiber for each fiber batch by measuring the divergence of the emitted radiation in the far field. Due to the wavelength dependence of the NAe2, this is done for several typical wavelengths in the working range of the fiber. 

An article on how measured fiber parameters help to choose the best coupling and collimation optics can be found here.

Nominal Numerical Aperture NA

Schäfter+Kirchhoff obtain fibers from different manufacturers.
with nco and ncl as the refractive indices of fiber core and cladding, respectively.
For a typical single-mode or a polarization-maintaining fiber, the nominal value is NA = 0.12
This NA specification corresponds to the Gaussian angle distribution at a 1 - 5 %-level.

Fiber Coupling


More details can be found here.


The collimated beam diameter Øbeam is a function of the ­collimating focal length f’ and the ­numerical aperture NA of the ­single-mode fiber.

{!{!{\emptyset_{beam}=2\cdot f\prime\cdot{NA}_{e^2}}!}!}
More details can be found here.