Engineering
Fiber-Optic Numerical Aperture (NA), Acceptance Angle & V-Number
Explore an ideal circular step-index fiber: numerical aperture, a ray-model acceptance half-angle and V-number, using consistent units and an approximate higher-mode cutoff.
An illustrative step-index model, not a commercial-fiber specification
The supplied indices and radius describe an illustrative circular step-index fiber. They are not a verified specification for SMF-28 or another commercial product. Use phase refractive indices at the operating wavelength; holding them fixed while changing wavelength is a simplified comparison.
NA=sqrt(ncore²-ncladding²). In the meridional-ray model, the external half-angle is asin(NA/nexternal). This example defaults to air with external index 1. The full cone angle is twice the half-angle. The formula is not a sharp coupling limit for a single-mode optical field.
V=2 pi a NA/lambda uses core radius a and vacuum wavelength lambda in the same units, here micrometres. For an ideal weakly guiding circular step-index fiber, approximately 2.405 is the higher-order-mode cutoff. The label is a model comparison, not a guarantee for every index profile.
Single spatial-mode operation does not mean dispersion-free transmission: chromatic and waveguide dispersion still matter. The example does not model polarization splitting, losses, bends or coupling efficiency.
Numerical aperture in D2
Core index must exceed cladding index before taking the square root.
=IF(COUNT(B2:B3)<>2,"Enter numeric refractive indices",IF(OR(B2<=0,B3<=0),"Indices must be positive",IF(B2<=B3,"Requires core index > cladding index",SQRT(B2^2-B3^2))))Ray-model acceptance half-angle in degrees, D3
B6 is the external refractive index. A ratio above 1 is reported as outside the chosen asin relation instead of being silently clamped.
=IF(NOT(ISNUMBER(D2)),D2,IF(COUNT(B6)<>1,"Enter numeric external index",IF(B6<=0,"External index must be positive",IF(D2/B6>1,"Ray-angle relation outside selected domain",DEGREES(ASIN(D2/B6))))))Unrounded V-number in D4
Retain full precision for the cutoff comparison.
=IF(NOT(ISNUMBER(D2)),D2,IF(COUNT(B4:B5)<>2,"Enter numeric radius and wavelength",IF(MIN(B4:B5)<=0,"Radius and wavelength must be positive",2*PI()*B4*D2/B5)))Approximate cutoff comparison in D5
Uses the unrounded V-number with the explicitly rounded reference 2.405.
=IF(NOT(ISNUMBER(D4)),D4,IF(D4<2.405,"Below approximate higher-mode cutoff","At or above approximate higher-mode cutoff"))Try the example
Use the original values or change them to test your understanding.