Bandwidth and manufacturing tolerances on multimode fiber refractive-index profiles Colin Pask

Australian National University, Research School of Phys­ical Sciences, Canberra, ACT 2600, Australia. Received 12 February 1981. 0003-6935/81/142339-02$00.50/0. © 1981 Optical Society of America.

In this Letter we discuss the required tolerances on fiber parameters with respect to variations along the fiber length.

Multimode fibers are usually fabricated using the power law profile design which leaves , the exponent; y(λ), the dis­persion parameter; and λ, the operating wavelength to be chosen. The choice of dopant fixes y and then theory predicts which should be used for a given λ and required bandwidth BW.1 Graphs of BW vs for fixed λ, or BW vs λ for fixed , show sharp maxima which remain, in somewhat less extreme form, when source bandwidth is included.1,2 Perturbations around the ideal profile and of varying amplitude along the fiber cause a similar bandwidth reduction, although the po­sition of the maximum remains unchanged (see Ref. 3 for computational results and Ref. 2 for possible experimental examples). Thus, the -y-λ optimization procedure is still useful in imperfect fibers giving reasonable working param­eters although overoptimistic estimations of BW. In this Letter we analyze the required tolerances on using analytical transit time results4 for = (z).

We assume that the spatial variations in are slow so that mode coupling is not present, and over a length L of fiber there will be variations in , which may be random or systematic (Fig. 1), of amplitude ε, where

The key result which we will apply is that as far as transit times are concerned, such a fiber behaves like a uniform fiber with = e, an equivalent exponent.4 This result includes the material dispersion effects of importance here. When ε is small a fiber varying around = o has4

Fig. 1. Possible variations, amplitude e of exponent over a length L of fiber: A, random variation; B, systematic linear decrease of ; C, systematic increase away from the fiber endpoints modeled by a

sine function.

Fig. 2. Bandwidth EW vs operating wavelength λ for fibers with exponent varying around 0 to give equivalent exponent e = o + δ. Curves are labeled by δ, which reflects the magnitude of the ex­ponent variations via Eq. (2). o= 1.91 and material properties

follow from Ref. 6. Small source spectral width is assumed.

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Fig. 3. Bandwidth BW of fibers having exponent varying about α0 = 1.91 vs δ [Eq. (2)]. Operating wavelength is optimum for α = αe = αo = 1.91, i.e., the δ = 0 uniform fiber. Source spectral width is not

taken into account.

where ƒ is the average of ƒ along the fiber. For the systematic variations in Fig. 1, δ = –ε/2 (steady decrease), δ = 2ε/π (symmetric changes about end points). For example, the measurements of French et al.5 suggest 0.08 and εf ≅ 0.04, or δ/α0 ≅ 2%, for their fibers. Equation (1) implies a maxi­mum variation in n — nclad of 100ε/(α + ε) s 50ε%, so that only small profile changes are involved.

In Fig. 2 we plot a 6-dB bandwidth2 vs wavelength for a fiber having α0 = 1.91 and germanium-phosphorus doping giving y(λ) as in Ref. 6. [For simplicity, we set Δ = 0.01, ignore variations in n and N, and use αopt

= 2 + y — 2.4Δ to obtain a working y(λ) from Fig. 7 of Ref. 6.] Variations of α along the fiber are taken into account theoretically by using α = αe in the bandwidth equations2 and lead to the displaced curves in Fig. 2. We observe that δ/α0 ≅ 2% leads to changes in the optimum operating wavelength of order 20% or more de­pending on the sign of 8, and this behavior is unchanged when source spectral width is included. To see the effects in a different way, we assume that the operating wavelength is chosen to be λopt, the wavelength for which α = 1.91 is opti­mum. The bandwidth for fiber variations around αo is given in Fig. 3, which indicates that small variations along the fiber of the power law exponent can lead to very large bandwidth reductions. The reductions will be still significant but a little less dramatic when a nonzero source spectral width is as­sumed.

We conclude that when fiber fabrication is based on the power law optimization scheme, the exponent α should be controlled as tightly as possible since large bandwidth re­ductions, or large changes in the desirable operating wave­length, may result from spatial variations in α along the fiber.

The financial support of Telecom Australia is gratefully acknowledged.

References 1. R. Olshansky and D. B. Keck, Appl. Opt. 15, 483 (1976). 2. M. Horiguichi, Y. Ohmori, and H. Takata, Appl. Opt. 19, 3159

(1980). 3. D. Marcuse, Appl. Opt. 18, 4003 (1979). 4. C. Pask, Opt. Quantum. Electron. 12, 281 (1980). 5. W. G. French, G. W. Tasker, and J. R. Simpson, Appl. Opt. 15,1803

(1976).

6. M. Horiguichi, Y. Ohmori, and T. Miya, Appl. Opt. 18, 2223 (1979).

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