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Chapter 12: Problem 29

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?

Short Answer

Expert verified
Answer: The pulse will spread to twice its initial width over a distance of approximately 0.33 km in the dispersive medium.

Step by step solution

01

Recall the pulse broadening formula for second-order dispersion

The pulse broadening due to second-order dispersion can be calculated using the formula: \(\Delta T = T \sqrt{1 + \frac{16}{T^2} (\beta_{2} z)\log(2)} \) where \(\Delta T\) is the new pulse width after propagation, \(T\) is the initial pulse width, \(\beta_{2}\) is the second-order dispersion parameter, \(z\) is the propagation distance, and \(\log(2)\) is the natural logarithm of 2.
02

Set the width after propagation to twice the initial width

Since we want to find the distance over which the pulse spreads to twice its initial width, we set \(\Delta T = 2T\): \( 2T = T \sqrt{1 + \frac{16}{T^2} (\beta_{2} z)\log(2)} \)
03

Solve for the propagation distance, z

To find the value of z, we first divide both sides of the equation by T and then square both sides: \( \frac{2T}{T} = \sqrt{1+\frac{16}{T^2} (\beta_{2} z)\log(2)} \) \( 2 = \sqrt{1+\frac{16}{T^2} (\beta_{2} z)\log(2)} \) Squaring both sides: \( 2^2 = 1+ \frac{16}{T^2} (\beta_{2} z)\log(2) \) Now, isolate the term \((\beta_{2} z)\): \( 4-1 = \frac{16}{T^2} (\beta_{2} z)\log(2) \) \( 3 = \frac{16}{T^2} (\beta_{2} z)\log(2) \) Now, solve for \(z\): \( z = \frac{3T^2}{16 \beta_{2}\log(2)} \)
04

Calculate the distance

With the values provided in the problem statement, \(T = 5\,\text{ps}\) and \(\beta_{2} = 10\,\text{ps}^2/\text{km}\), we can calculate the distance over which the pulse width doubles. \( z = \frac{3(5\,\text{ps})^2}{16 (10\,\text{ps}^2/\text{km})\log(2)} \) \( z \approx 0.33\,\text{km} \) Hence, the pulse will spread to twice its initial width over a distance of approximately \(0.33\,\text{km}\) in the dispersive medium.

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Most popular questions from this chapter

Over a small wavelength range, the refractive index of a certain material varies approximately linearly with wavelength as \(n(\lambda) \doteq n_{a}+n_{b}\left(\lambda-\lambda_{a}\right)\), 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.

The plane \(z=0\) defines the boundary between two dielectrics. For \(z<0\), \(\epsilon_{r 1}=9, \epsilon_{r 1}^{\prime \prime}=0\), and \(\mu_{1}=\mu_{0} .\) For \(z>0, \epsilon_{r 2}^{\prime}=3, \epsilon_{r 2}^{\prime \prime}=0\), and \(\mu_{2}=\mu_{0}\) Let \(E_{x 1}^{+}=10 \cos (\omega t-15 z) \mathrm{V} / \mathrm{m}\) and find \((a) \omega ;(b)\left\langle\mathbf{S}_{1}^{+}\right\rangle ;(c)\left\langle\mathbf{S}_{1}^{-}\right\rangle\); (d) \(\left\langle\mathbf{S}_{2}^{+}\right\rangle\).

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 wave starts at point \(a\), propagates \(1 \mathrm{~m}\) through a lossy dielectric rated at \(0.1 \mathrm{~dB} / \mathrm{cm}\), reflects at normal incidence at a boundary at which \(\Gamma=0.3+j 0.4\), and then returns to point \(a .\) Calculate the ratio of the final power to the incident power after this round trip, and specify the overall loss in decibels.

A uniform plane wave in region 1 is normally incident on the planar boundary separating regions 1 and 2. If \(\epsilon_{1}^{\prime \prime}=\epsilon_{2}^{\prime \prime}=0\), while \(\epsilon_{r 1}^{\prime}=\mu_{r 1}^{3}\) and \(\epsilon_{r 2}^{\prime}=\mu_{r 2}^{3}\), find the ratio \(\epsilon_{r 2}^{\prime} / \epsilon_{r 1}^{\prime}\) if \(20 \%\) of the energy in the incident wave is reflected at the boundary. There are two possible answers.
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