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Catalog > Matching Numerical Apertures

Matching Numerical Apertures

In optical fiber use, it is extremely important to attempt to match mating fiber NA’s. With the same fiber core diameter it is usually best to center the two fiber ends using a tapered sleeve or set of mating connectors. If there is a large mismatch in the fiber diameters, a tapered optical fiber may be a good solution. Two different diameter spheres in a coupler may also achieve the needed result.

The target is to match the NA’s as best possible since the light loss in a lens coupling will vary as the square of the NA. You should also take care that the NA’s of two coupled fibers are matched. Optical loss due to fiber NA mismatch is represented by:

Where NA₁ is transmitting fiber and NA₂ is the receiving fiber. If NA₁ is less than NA₂ then the mismatch loss is zero. When there is a difference in NA between the two fibers, the higher NA should be the second in line so that all or most of the flux from the first fiber is accepted by the second.

The total flux lost will be a function of how well the input acceptance cone of the second fiber is filled (without overfilling) by the output source fiber. These fiber-to-fiber matches are also a function of other conditions besides NA, such as core alignment, angle, distance between the fibers, surface cuts, angles and finishes, as well as any optical coupling media characteristics.

When using optical fiber in an optical system, the acceptance cones should match at each interface. The F-number, F#, of a lens is given by:

where efl is the lens effective focal length and D is the diameter of its entrance pupil. The relation between the F# and the numerical aperture (NA) is:

So in order to match NAs between the lens system and the optical fiber, we need:

This means, if we have a lens with an efl of 50mm and a 18mm aperture (F# = 2.8), we would need an optical fiber with an NA of at least 0.18 to get an ideal match. This does not include any losses due to Fresnel reflections and lens focal spot size versus core size mismatches.

Figure 2-12

Because of the rapid change of the refractive index of quartz in the UV (Figure 2-12), great care should be taken when designing spectrometers or other systems that must work in both the visible and the UV. Either focus adjustments need to be included or a compromise focus position must be chosen. Since the focal length decreases with wavelength (about 13% from 550nm to 220nm), the NA increases. Matching the NAs and assuring the focus spot is just inside the fiber core at the shortest wavelength would be the best situation. The balance between these two characteristics may still leave some losses.

Figure 2-12

The efl and NA characteristics for a fused quartz plano-convex, singlet lens versus wavelength can be seen in Figure 2-13.

Figure 2-13

This axial chromatic aberration problem is solved in visible and IR optical systems, to some extent, by adding color correction using optical elements with complementary characteristics. In the UV, there are very few choices of materials to obtain color correction. The refractive index versus wavelength curve in the graph illustrates the reason for some of the problem.

A generalized illustration of lens coupling into an optical fiber for two different wavelengths may help to clarify the phenomena of UV chromatic aberrations in quartz. As can be seen in the illustration, not only is the focal length shorter at 220nm (~13% less), but the exit cone angle is larger. Both of these changes from the 550nm case can cause losses when trying to use optical fiber over an extended wavelength range.

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