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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.

Short Answer

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Answer: For total transmission, the arrangement is: 1. Dielectric slab 2. λ/4 airspace 3. Dielectric slab 4. λ/4 airspace 5. Dielectric slab 6. λ/4 airspace 7. Dielectric slab For highest reflectivity, the arrangement is: 1. Dielectric slab 2. λ/2 airspace 3. Dielectric slab 4. λ/2 airspace 5. Dielectric slab 6. λ/2 airspace 7. Dielectric slab

Step by step solution

01

Finding the Transmission Coefficient

To determine the transmission coefficient of an arrangement, we first find the transmission coefficients of the individual dielectric-air interfaces and air-dielectric interfaces. Since the boundary conditions don't change, the transmission coefficients are the same for each interface. Therefore, we can find the overall transmission coefficient as the product of the transmission coefficients of the individual interfaces. We know that for a normally incident wave, the transmission coefficient, \(T\), is given by $$ T = \frac{2\eta_{2}}{\eta_{1} + \eta_{2}}, $$ where \(\eta_{1}\) and \(\eta_{2}\) are the intrinsic impedances of the first and second media, respectively. For total transmission, we want the overall transmission coefficient to be equal to 1. A set of four slabs and airspaces should be connected to allow this to happen.
02

Arrangement for Total Transmission

To achieve total transmission through the stack, we'll arrange the four dielectric slabs with quarter-wavelength airspaces between them: 1. Dielectric slab 2. \(\lambda/4\) airspace 3. Dielectric slab 4. \(\lambda/4\) airspace 5. Dielectric slab 6. \(\lambda/4\) airspace 7. Dielectric slab This arrangement helps to cancel out the reflections that occur at each interface, such that the transmitted wave builds up constructively, and thus all the wave energy passes through the stack.
03

Arrangement for Highest Reflectivity

Now, we are interested in an arrangement that presents the highest reflectivity for the incident wave. In this case, we want the magnitude of the overall reflection coefficient to be maximum. For the highest reflectivity, we'll arrange the dielectric slabs with half-wavelength airspaces between them: 1. Dielectric slab 2. \(\lambda/2\) airspace 3. Dielectric slab 4. \(\lambda/2\) airspace 5. Dielectric slab 6. \(\lambda/2\) airspace 7. Dielectric slab This arrangement leads to reflections that add up constructively and result in the maximum reflectivity of the stack.

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Most popular questions from this chapter

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.

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.

Suppose that \(\phi\) 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 \(n_{2}\).

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.

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?

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