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Chapter 13: Problem 28

In a symmetric slab waveguide, \(n_{1}=1.50, n_{2}=1.45\), and \(d=10 \mu \mathrm{m}\). (a) What is the phase velocity of the \(m=1 \mathrm{TE}\) or \(\mathrm{TM}\) mode at cutoff? (b) How will your part ( \(a\) ) result change for higher-order modes (if at all)?

Short Answer

Expert verified
Answer: To calculate the phase velocity at the cutoff for m=1 TE or TM modes in a symmetric slab waveguide, we first must find the propagation constants and cutoff conditions for both modes. At the cutoff condition, TE and TM modes will have the same phase velocity. We can then use the calculated effective refractive index to find the phase velocity using the equation \(v_{\text{p}} = \dfrac{c}{n_{\text{eff}}}\), where \(c\) is the speed of light. Higher-order modes will have different frequency cutoff conditions and effective refractive indices but will generally not change the phase velocity significantly for a given frequency, as the influence of the waveguide becomes less dominant.

Step by step solution

01

(Step 1: Find the propagation constants for the waveguide)

(For a symmetric slab waveguide, we need to find the propagation constants for both TE and TM modes. In both cases, the propagation constant can be expressed as \(\beta = k n_{\text{eff}}\), where \(k\) is the free space wavenumber and \(n_{\text{eff}}\) is the effective refractive index.)
02

(Step 2: Find the cutoff conditions)

(For the m=1 TE or TM mode to propagate, we need to satisfy the cutoff conditions. The cutoff conditions for TE and TM modes in a symmetric slab waveguide are given by: TE mode: \(\text{tan}(\delta) = \dfrac{n_{2}^{2}}{n_{1}^{2} - n_{2}^{2}}\times \dfrac{k d}{2}\) TM mode: \(\text{tan}(\delta) = \dfrac{n_{1}^{2} n_{2}^{2}}{n_{1}^{2} - n_{2}^{2}}\times \dfrac{k d}{2}\) Where \(\delta = \pi \dfrac{m}{2}\) for the m-th mode and \(d\) is the thickness of the waveguide.)
03

(Step 3: Calculate phase velocity at cutoff for m=1 TE and TM mode)

(At the cutoff condition, TE and TM modes will have the same phase velocity. The phase velocity \(v_{\text{p}}\) can be calculated using: \(v_{\text{p}} = \dfrac{c}{n_{\text{eff}}}\), where \(c\) is the speed of light. Solve the above equations for \(n_\text{eff}\) at cutoff, and then use this value to calculate the phase velocity \(v_{\text{p}}\).)
04

(Step 4: Analyze higher-order modes)

(Higher-order modes will have different frequency cutoff conditions and effective refractive indices. We can go back to Step 2 and replace the mode number \(m\) with higher-order mode numbers to find the respective cutoff and propagation constants, and hence their phase velocities. Generally, as we go to higher-order modes, the phase velocity will not change significantly for a given frequency because the influence of the waveguide becomes less dominant.)
05

(Summary)

(To recap, we first found the propagation constants for TE and TM modes in a symmetric slab waveguide, calculated the cutoff conditions, and then found the phase velocity at cutoff for the m=1 TE and TM modes. We also briefly discussed the impact of higher-order modes on the results.)

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Most popular questions from this chapter

A transmission line constructed from perfect conductors and an air dielectric is to have a maximum dimension of \(8 \mathrm{~mm}\) for its cross section. The line is to be used at high frequencies. Specify the dimensions if it is (a) a two-wire line with \(Z_{0}=300 \Omega ; (b)\) a planar line with \(Z_{0}=15 \Omega\); (c) a \(72 \Omega\) coax having a zero-thickness outer conductor.

A symmetric dielectric slab waveguide has a slab thickness \(d=10 \mu \mathrm{m}\), with \(n_{1}=1.48\) and \(n_{2}=1.45\). If the operating wavelength is \(\lambda=1.3 \mu \mathrm{m}\), what modes will propagate?

The conductors of a coaxial transmission line are copper \(\left(\sigma_{c}=5.8 \times\right.\) \(\left.10^{7} \mathrm{~S} / \mathrm{m}\right)\), and the dielectric is polyethylene \(\left(\epsilon_{r}^{\prime}=2.26, \sigma / \omega \epsilon^{\prime}=0.0002\right) .\) If the inner radius of the outer conductor is \(4 \mathrm{~mm}\), find the radius of the inner conductor so that \((a) Z_{0}=50 \Omega ;(b) C=100 \mathrm{pF} / \mathrm{m} ;(c) L=0.2 \mu \mathrm{H} / \mathrm{m}\). A lossless line can be assumed.

Pertinent dimensions for the transmission line shown in Figure \(13.2\) are \(b=\) \(3 \mathrm{~mm}\) and \(d=0.2 \mathrm{~mm}\). The conductors and the dielectric are nonmagnetic. (a) If the characteristic impedance of the line is \(15 \Omega\), find \(\epsilon_{r}^{\prime}\). Assume a low-loss dielectric. ( \(b\) ) Assume copper conductors and operation at \(2 \times 10^{8}\) \(\mathrm{rad} / \mathrm{s}\). If \(R C=G L\), determine the loss tangent of the dielectric.

A lossless parallel-plate waveguide is known to propagate the \(m=2 \mathrm{TE}\) and TM modes at frequencies as low as \(10 \mathrm{GHz}\). If the plate separation is \(1 \mathrm{~cm}\), determine the dielectric constant of the medium between plates.
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