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Example: Calculation of the magnification and transmission on Tapered Fibers.

Tapered Fibers or Tapers have a wide range of applications in spectroscopy, sensing or laser optics. They are produced by changing the drawing speed during the fiber drawing process such that the fiber has different core diameters d₁ and d₂ on each side. The taper is the region with the length L in which the core diamter decreases linearly.

A taper is an economical means of magnifying or reducing a radiation field. Depending on the application light may be launched into the taper from the side with higher core diameter (down-taper) or from the opposite side (up-taper). The magnification M is given by the equation

It is important to note that the beam quality on both taper sides is preserved. A beam with a beam diameter d₁ and far-field angle θ_i launched into the down-taper will transform into a beam with beam diameter d₂ an far-field angle θ₀ = (M · θ₁) As only rays with angles smaller than the numerical aperture NA of the fiber are guided within the fiber, the launching condition for the down-taper reads

For illustration the drawing shows 2 different rays within a down-taper launched at different angles. At each reflection the reflection angel φ decreases and rays with θ_i > NA / M will be radiated out of the core. If the launching condition is not met the radiation which is coupled out of the taper leads to considerable power loss. To quantify the radiation loss of a down-taper let us consider a beam with a beam diameter d₁ and a far-field angle equal to the numerical aperture NA of the fiber. In this case of a completely filled fiber core the taper transmission T will be

Example:

For d₁ = 800 µm and d₂ = 400 µm the magnification is M = 2 and the transmission will be T = 0.25.

The taper transmission and the launching conditions are independent of the taper length L. Increasing L increases the number of reflections at the core-cladding interface. This may be an advantage for certain applications. For example, longer tapers distribute dissipated radiation over a longer taper length, reducing the power density within the fiber coating.

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