Chapter 12

A uniform plane wave in free space is normally incident onto a dense dielectric plate of thickness $\lambda /4$, having refractive index $n$. Find the required value of $n$ such that exactly half the incident power is reflected (and half transmitted). Remember that $n>1$.

A uniform plane wave is normally incident onto a slab of glass $(n=1.45)$ whose back surface is in contact with a perfect conductor. Determine the reflective phase shift at the front surface of the glass if the glass thickness is (a) $\lambda /2;(b)\lambda /4;(c)\lambda /8$.

You are given four slabs of lossless dielectric, all with the same intrinsic impedance, $\eta$, known to be different from that of free space. The thickness of each slab is $\lambda /4$, where $\lambda$ is the wavelength as measured in the slab material. The slabs are to be positioned parallel to one another, and the combination lies in the path of a uniform plane wave, normally incident. The slabs are to be arranged such that the airspaces between them are either zero, one-quarter wavelength, or one-half wavelength in thickness. Specify an arrangement of slabs and airspaces such that $(a)$ the wave is totally transmitted through the stack, and $(b)$ the stack presents the highest reflectivity to the incident wave. Several answers may exist.

Suppose that $\varphi$ in Figure $12.17$ is Brewster's angle, and that ${\theta }_1$ is the critical angle. Find $n_0$ in terms of $n_1$ and $n_2$.

Show how a single block of glass can be used to turn a p-polarized beam of light through ${180}^{\circ }$, with the light suffering (in principle) zero reflective loss. The light is incident from air, and the returning beam (also in air) may be displaced sideways from the incident beam. Specify all pertinent angles and use $n=1.45$ for glass. More than one design is possible here.

Over a small wavelength range, the refractive index of a certain material varies approximately linearly with wavelength as $n(\lambda )\doteq n_a+n_b(\lambda -{\lambda }_a)$, where $n_a,n_b$ and ${\lambda }_a$ are constants, and where $\lambda$ is the free-space wavelength. (a) Show that $d/d\omega =-\left(2\pi c/{\omega }^2\right)d/d\lambda$. (b) Using $\beta (\lambda )=2\pi n/\lambda$, determine the wavelength- dependent (or independent) group delay over a unit distance. ( $c$ ) Determine ${\beta }_2$ from your result of part $(b).(d)$ Discuss the implications of these results, if any, on pulse broadening.

A $T=5$ ps transform-limited pulse propagates in a dispersive medium for which ${\beta }_2=10{\mathrm{ps}}^2/\mathrm{km}$. Over what distance will the pulse spread to twice its initial width?

A $T=20$ ps transform-limited pulse propagates through $10\mathrm{km}$ of a dispersive medium for which ${\beta }_2=12{\mathrm{ps}}^2/\mathrm{km}$. The pulse then propagates through a second $10\mathrm{km}$ medium for which ${\beta }_2=-12{\mathrm{ps}}^2/\mathrm{km}$. Describe the pulse at the output of the second medium and give a physical explanation for what happened.