Given a general elliptically polarized wave as per Eq. (93): $$\mathbf{E}_{s}=\left[E_{x 0} \mathbf{a}_{x}+E_{y 0} e^{j \phi} \mathbf{a}_{y}\right] e^{-j \beta z}$$ (a) Show, using methods similar to those of Example 11.7, that a linearly polarized wave results when superimposing the given field and a phaseshifted field of the form: $$\mathbf{E}_{s}=\left[E_{x 0} \mathbf{a}_{x}+E_{y 0} e^{-j \phi} \mathbf{a}_{y}\right] e^{-j \beta z} e^{j \delta}$$ where \(\delta\) is a constant. \((b)\) Find \(\delta\) in terms of \(\phi\) such that the resultant wave is linearly polarized along \(x\).

Understanding the superposition principle is essential when studying waves and their interactions. In physics, the superposition principle states that when two or more waves overlap at a point, the resultant wave is the sum of the individual waves at that point. This concept applies to all linear systems, and electromagnetic fields are no exception.

For instance, if you have an elliptically polarized wave, like the one given in the exercise, and you superimpose it with a phase-shifted wave, the resulting field is simply the vector sum of the two fields at each point in space. Mathematically, this is demonstrated by adding corresponding components of the two waves to get a resultant wave equation. This approach is often used to simplify complex wave behaviors and predict the outcome when waves interact, such as creating linear polarization from elliptical polarization through phase-shifting.

Linear polarization is a particular state of polarization where the electric field vector of the wave oscillates in a single plane along the direction of propagation. The simplified definition makes it an excellent illustration for understanding wave interactions under the superposition principle. In the context of the exercise, achieving linear polarization involves the manipulation of two elliptically polarized waves.

When two such waves are correctly aligned through superposition and exhibit a specific phase relationship, a scenario can emerge where their combined electric fields cancel out in one axis, resulting in oscillation only along the other axis. The precise manipulation of this polarization state is fundamental in various optical technologies, such as filters and modulators, where control over the light's polarization can enhance or permit different functionalities.

The phase shift is a crucial concept when dealing with wave interference and the formation of new wave patterns. It represents the angle by which one wave is offset relative to another wave. A phase shift can alter the characteristics of a wave, such as transforming an elliptically polarized wave into a linearly polarized one.

In this exercise, we saw that by introducing a phase shift to one of the waves and then superimposing it upon the original wave, we could affect the polarization of the resulting wave. To obtain linear polarization, specifically along the x-axis, it's necessary to calculate the correct phase shift such that the y-components of the waves cancel each other out, leaving only the x-component. By finding the right phase shift, we control the wave's interference pattern, a concept widely used in communication systems, optics, and even quantum mechanics.