Suppose \(\epsilon_{r x}=2.15, \epsilon_{r y}=2.10\), and the wave electric field at input is polarized at \(45^{\circ}\) to the positive \(x\) and \(y\) axes. (a) Determine, in terms of the free space wavelength, \(\lambda\), the shortest length of the material, such that the wave, as it emerges from the output, is circularly polarized. ( \(b\) ) Will the output wave be right or left circularly polarized? Problem \(11.30\) is good background. Suppose that the length of the medium of Problem \(11.31\) is made to be twice that determined in the problem. Describe the polarization of the output wave in this case.

Phase difference is a fundamental concept in the study of waves, referring to the relative timing between two waves or wave components at a given point in space. In the context of electromagnetic waves, which include light, phase difference can be critical in determining how the waves interact and what type of polarization they exhibit.

When two wave components, such as the x and y components of an electromagnetic wave, have a certain phase difference, they can combine to form various types of polarization. In the provided exercise, the phase difference required for circular polarization was \(\frac{\pi}{2}\). This is a quarter of a full wave cycle, and when the electric field components of a wave are out of phase by this amount, each component reaches its peak at different times, resulting in a circular motion of the electric field vector.

For the electromagnetic wave to be circularly polarized after passing through a material, there must be exactly this phase difference between the components upon exit from the material. By careful manipulation of the wave's properties and the material through which it travels, one can control the resulting polarization—a critical application in fields such as optics and telecommunications.

Circular polarization occurs when two perpendicular electromagnetic wave components have equal amplitudes and a phase difference of \(\frac{\pi}{2}\). The electric field of such a wave rotates in a circular motion as the wave propagates.

Electromagnetic waves can be polarized in different ways, the simplest being linear polarization, where the electric field oscillates in a straight line. However, if the fields oscillate asynchronously, they can produce elliptical or circular polarization. If viewed along the direction of the wave, and the rotation is clockwise, it is called right-handed circular polarization, as it was found in the provided solution. If it rotates counterclockwise, it is referred to as left-handed circular polarization.

The behavior of materials under different polarizations is critical for designing optical systems. For example, 3D glasses work by only allowing one circular polarization in each lens, hence creating a stereoscopic effect. Recognizing the precise form of polarization in an electromagnetic wave and the conditions that lead to it is essential for harnessing and applying the wave's properties in applications like satellite communication and molecular spectroscopy.

Relative permittivity, often symbolized as \(\epsilon_r\), is a measure of a material's ability to polarize in response to an applied electric field, which affects how electromagnetic waves propagate through the material. It relates to the easier-to-grasp concept of electrical 'kinkiness' or resistance to forming an electric field within a material.

In simpler terms, it defines how much the material reduces the speed of light compared to its speed in a vacuum. The relative permittivity is a dimensionless quantity and is often used in the context of dielectric materials in capacitors or insulators. In the exercise provided, different relative permittivities for the x and y axes (\(\epsilon_{r x}=2.15, \epsilon_{r y}=2.10\)) are given. This anisotropy can lead to a phase difference building up as the wave traverses the material, which is crucial for creating the precise polarization state of the wave.

Understanding the concept of relative permittivity is critical when designing materials for electronics, creating insulating components, or even explaining natural phenomena such as how light bends when entering water or another transparent medium, all of which impact the world of optical and electronic engineering.