Optical Fibers

Abstract

This chapter describes the waveguiding properties of optical fibers, with emphasis on the single-mode fibers used in optical communications. After examining modal behavior and attenuation processes, the dispersive propagation of ultrashort pulses is treated in some detail. A section is dedicated to the description of several different types of fibers and their properties, followed by a presentation of some fiber components, such as fiber couplers and fiber mirrors, which play an important role in the setup of fiber sensors and active fiber devices. The last two sections specifically deal with the active devices, such as fiber amplifiers and fiber lasers.

Problems

6.1

A step-index fiber has a numerical aperture of 0.15, a core refractive index of 1.47 and a core diameter of $20~{\upmu }$m. Calculate (i) the maximum acceptance angle of the fiber, (ii) the refractive index of the cladding, (iii) the approximate number of modes with a wavelength of 800 nm that the fiber can carry.

6.2

An optical power of 10 mW is launched into an optical fiber of length 500 m. If the power emerging from the other end is 8 mW, calculate the fiber attenuation $\alpha _o$ in dB/km.

6.3

An erbium-doped fiber amplifier has length $L=11$ m. If the stimulated-emission cross-section at $\lambda =1530$ nm is $1.2 \times 10^{-20}$ cm$^2$ and the population inversion is $N_2 - N_1= 7 \times 10^{17}$ cm$^{-3}$, calculate the small-signal gain in dB units.

6.4

A light beam at $\lambda =1550$ nm is coupled into a step-index optical fiber having numerical aperture $NA = 0.12$ and core radius $a = 4.5~{\upmu }$m. (i) specify whether the fiber behavior will be monomodal or multimodal; (ii) calculate the parameter w of the fundamental mode.

6.5

A transform-limited Gaussian laser pulse having duration $\tau _p=4$ ps and angular frequency $\omega _o$ is coupled into a single-mode optical fiber. The fiber has an attenuation coefficient $\alpha _o = 0.15$ dB/km and a frequency-dependent propagation constant $\beta (\omega )$ following the law:

$$\begin{aligned} \beta (\omega ) = \beta _o + \beta _1(\omega - \omega _o) + (1/2)\beta _2(\omega - \omega _o)^2 \end{aligned}$$

(6.39)

where $\beta _1 = 4.90 \times 10^{-9}$ s/m, $\beta _2 = - 1 \times 10^{-27}$ s$^2$/m. By assuming that the fiber length is $L=50$ km, calculate: (i) the time taken by the peak of the pulse to propagate along the fiber; (ii) the pulse duration $\tau _p(L)$ at the fiber output; (iii) the ratio r between the output and input energy of the laser pulse; (iv) the ratio $r' = P_{L}/P_{0}$, where $P_{0}$ and $P_{L}$ are the input and output peak power of the laser pulse, respectively.

6.6

Consider a step-index optical fiber with a core radius $a = 4.5~{\upmu }$m and core refractive index $n_1 = 1.494$. Assuming that the normalized frequency is $V=2.15$ at $\lambda =1550$ nm, calculate: (i) the cladding refractive index; (ii) the width w of the fundamental mode; (iii) the wavelength at which the fiber ceases to be single-mode.

6.7

An erbium-doped fiber amplifier has length $L = 10$ m and a stimulated emission cross section $\sigma =1.2 \times 10^{-20}$ cm$^2$ at $\lambda =1550$ nm. Calculate the value of the population inversion $N_2-N_1$ required to attain a small signal gain of 20 dB.

6.8

Calculate pitch and length of a fiber Bragg grating having a reflectivity peak $R_\mathrm{{max}}=0.995$ at $\lambda = 1500$ nm, assuming that the core average refractive index is $n=1.5$ and that ${\varDelta }n = 2 \times 10^{-4}$.

6.9

Calculate the effective refractive index $n_{eff}$ at $\lambda =1550$ nm for a single-mode optical fiber having a core radius $a=5~{\upmu }$m, core refractive index $n_1=1.484$, cladding refractive index $n_2=1.480$, by assuming that $n_{eff}=n_{1}f_1+n_{2}f_2$, where $f_1$ and $f_2$ are the power fractions traveling inside the core and the cladding, respectively. The profile of the single mode is given by the Gaussian function $\mathrm{{exp}}[(x^2+y^2)/w^2]$. The value of $n_{eff}$ should be given with 4 digits.

6.10

A transform-limited Gaussian laser pulse having duration $\tau _p=1$ ps is coupled into a single-mode optical fiber with a dispersion parameter $\beta _2=10$ ps$^2$/km. Calculate at which propagation distance the pulse duration is doubled.