Problems

Problem 10.1 Make a mindmap of the material covered in this chapter. For more information about mindmaps, see Wikipedia

Problem 10.2 Describe the (possible) relevance of fiber optics in your line of work.

Problem 10.3 Explain why for optical fiber communication lines the wavelength of choice are 1310 and $1550~\text{nm}$.

Problem 10.4 Show that the phase factor $(m-1)\pi$ in eq:fiber:down-propagating-wave indeed leads to a vanishing $E$-field at the mirrors. Is this the only possible solution?

Problem 10.5 A plastic fiber has a core refractive index of $n_1=1.49$ and a cladding refractive index of $n_2=1.38$. Its core diameter is $1.00~\text{mm}$ and light with a wavelength of $633~\text{nm}$ is coupled into the fiber from air ($n_{\text{e}}=1.00$). Calculate: (a) the internal critical angle $\theta_{\text{i,c}}$ (b) the (external) critical angle $\bar{\theta}_{\text{e,c}}$ (c) the $\Delta$-parameter (Can you use the approximation?) (d) the numerical aperture (e) the $V$-number (Is this a singlemode or multimode fiber?) (f) the cut-off wavelength

Problem 10.6 Describe, in your own words, the effect of dispersion on a short pulse. What is the maximum length of fiber of a loss-less datalink, given that the dispersion parameter of the fiber $D=20~\text{ps/km}\cdot \text{nm}$ and the laser used to communicate has a spectral width of $1.0~\text{nm}$? The laser can send light pulses at a rate of $10~\text{GHz}$.

Problem 10.7 What is the maximum length of fiber of a dispersion-less datalink, given that the loss of the fiber $\alpha_{\text{dB}}=0.30~\text{dB/km}$ and a light pulse can no longer be discriminated from the background noise if $99\%$ of the light is lost?

Problem 10.8 Estimate the loss (in dB) due to the following situations in which two single mode fibers are coupled incorrectly:

(a) a fiber with a core diameter of $7.0~\mu\text{m}$ is coupled to a fiber with core diameter $6.0~\mu\text{m}$ (see figFiberCouplingLoss).

(b) a $500~\mu\text{m}$-air gap exists in between two fibers (see figFiberCouplingLoss). Both have a numerical aperture equal to 0.12 and a core diameter of $6.0~\mu\text{m}$.

To make your estimation, neglect reflection due to refractive index mismatch and assume the incoming light has a Gaussian beam profile with intensity

Here, $r$ is the radial coordinate in the $(x,y)$-plane (see figFiberTIR). Integrate over the fiber core into which the light is coupled and divide by $I_0$.

Note that these calculations are simplified, but they give a rough estimate of coupling losses.