Problems

Problem 10.1 Make a mind map of the material covered in this chapter. For more information about mind maps, see Wikipedia

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

Problem 10.3 For a conventional single-mode silica communication fiber, compare operation near 1310 nm and 1550 nm in terms of chromatic dispersion and attenuation. Why might a system designer choose either band?

Problem 10.4 Show that the phase factor $(m-1)\pi$ in Eq. (7) 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 incidence angle $\theta_{\text{i,c}}$, measured from the normal to the core–cladding interface (b) the maximum external acceptance angle $\bar{\theta}_{\text{e,c}}$, measured from the fiber axis (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 cutoff wavelength of the first higher-order mode; does the fundamental mode also disappear at this wavelength?

Problem 10.6 Describe, in your own words, the effect of dispersion on a short pulse. For a simplified loss-free link, assume the initial pulse duration is negligible and require the dispersion broadening $|D|L\Delta\lambda$ to remain below one bit period. Find the maximum length when $D=20~\text{ps}/(\text{km}\cdot\text{nm})$, $\Delta\lambda=1.0~\text{nm}$, and the bit rate is $10~\text{GHz}$.

Problem 10.7 For a simplified dispersion-free link with no connector or splice losses, find the maximum length if $\alpha_{\text{dB}}=0.30~\text{dB/km}$ and the receiver requires at least $1\%$ of the launched power.

Problem 10.8 Estimate two simplified coupling losses. In each part, report the transmitted fraction $\eta$ and loss $-10\log_{10}\eta$ in dB. Neglect Fresnel reflections, alignment error, and polarization mismatch.

(a) Two centered single-mode fibers have core diameters of 7.0 and $6.0~\mu\mathrm m$. Approximate their guided field amplitudes by coaxial Gaussian profiles with intensity radii $w_1=3.5~\mu\mathrm m$ and $w_2=3.0~\mu\mathrm m$. Use the normalized mode-overlap estimate

Why is core diameter alone insufficient for an exact coupling prediction?

(b) Two identical fibers with core radius $a=3.0~\mu\mathrm m$ are separated by a $500~\mu\mathrm m$ air gap. For this rough geometric collection model, let the emerging Gaussian intensity radius be $w_0=3.0~\mu\mathrm m$ and use $w(z)=\sqrt{w_0^2+(\mathrm{NA}\,z)^2}$ with $\mathrm{NA}=0.12$. Estimate the fraction of power inside the receiving core using

Explain why this geometric fraction is not an exact guided-mode coupling efficiency.