Consider total reflection at an interface between two nonmagnetic media, with relative refractive index \(n=c_{1} / c_{2}<1\). For angles of incidence \(\theta_{1}\) exceeding the critical angle of (8.6.42), Snell's law gives $$ \sin \theta_{2}=\frac{\sin \theta_{1}}{n}>1, $$ which implies that \(\theta_{2}\) is a complex angle with an imaginary cosine, $$ \cos \theta_{2}=\left(1-\sin ^{2} \theta_{2}\right)^{1 / 2}=j\left(\frac{\sin ^{2} \theta_{1}}{n^{2}}-1\right)^{1 / 2} $$ Substitute these relations in the case I reflection coefficient (8.6.28) to establish $$ R_{\mathbf{E} \perp}=e^{-i 2 \phi_{\perp}}, $$ where $$ \tan \phi_{\perp}=\frac{\left(\sin ^{2} \theta_{1}-n^{2}\right)^{1 / x}}{\cos \theta_{1}} $$ That is, the magnitude of the reflection coefficient is unity, but the phase of the reflected wave depends upon angle. Similarly show for case II from (8.6.36), that \(R_{\text {III }}=e^{-\text {jod with }}\) $$ \tan \phi \|=\frac{\left(\sin ^{2} \theta_{1}-n^{2}\right)^{1 / 2}}{n^{2} \cos \theta_{1}}=\frac{1}{n^{2}} \tan \phi_{\perp} . $$ Note that the two phase shifts are different, so that in general the state of polarization of an incident wave is altered.

Step by step solution

01

Calculate sin𝜃2 and cos𝜃2

We are given Snell's law for this situation as: $$ \sin \theta_{2}=\frac{\sin \theta_{1}}{n}>1, $$ As sin𝜃2 > 1, we deduce that 𝜃2 is a complex angle. Therefore, we need to find the imaginary part of cos𝜃2 using the relation: $$ \cos \theta_{2}=\left(1-\sin ^{2} \theta_{2}\right)^{1 / 2}=j\left(\frac{\sin ^{2} \theta_{1}}{n^{2}}-1\right)^{1 / 2} $$

02

Substitute sin𝜃2 and cos𝜃2 in the reflection coefficient equation

The given reflection coefficient equation is (8.6.28): $$ R_{\mathbf{E} \perp}=e^{-i 2 \phi_{\perp}}, $$ We have to establish the relation for 𝜃2 and 𝜑⊥, which is given by: $$ \tan \phi_{\perp}=\frac{\left(\sin ^{2} \theta_{1}-n^{2}\right)^{1 / 2}}{\cos \theta_{1}} $$

03

Derive the reflection coefficient for the Case II

Derive the required expression for Case II from (8.6.36): $$ R_{\text {III }}=e^{-\text {jod with }} $$ We have to find the relationship for 𝜑|| using the given equation: $$ \tan \phi \|=\frac{\left(\sin ^{2} \theta_{1}-n^{2}\right)^{1 / 2}}{n^{2} \cos \theta_{1}}=\frac{1}{n^{2}} \tan \phi_{\perp} . $$

04

Analyze the change in polarization

The phase shifts for both cases are different: $$ \phi_{\perp} \neq \phi_{\|} $$ As a result, the state of polarization of an incident wave will be altered as it undergoes total reflection between the two nonmagnetic media.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Snell's Law

Snell's Law is a fundamental principle in optics that describes how light waves bend, or refract, as they pass from one medium into another with a different refractive index. More formally, it gives us a relationship between the angle of incidence (\theta_{1}) and the angle of refraction (\theta_{2}). The law is mathematically written as:
\[\begin{equation}\sin\theta_{2} = \frac{n_1}{n_2} \sin\theta_{1}\end{equation}\]
where \(n_1\) and \(n_2\) are the refractive indices of the first and second media, respectively. When light travels from a medium with a higher refractive index to a lower one, and the angle of incidence exceeds the so-called critical angle, total internal reflection occurs. This means that all the light is reflected back into the original medium, a concept we explore further in the problem at hand.

Refractive Index

The refractive index is a measure that indicates how much a material slows down light compared to the speed of light in a vacuum. It is defined by the equation:
\[\begin{equation}n = \frac{c}{v}\end{equation}\]
where \(c\) is the speed of light in a vacuum, and \(v\) is the speed of light in the material. The refractive index is crucial in determining various optical properties, such as the critical angle for total internal reflection, which in turn plays a significant role in understanding phenomena like fiber optics and the shimmering effect known as a mirage.

Reflection Coefficient

When it comes to understanding how waves interact with surfaces, the reflection coefficient is a critical term. It describes the amplitude of the reflected wave in relation to the incident wave at an interface. For a wave encountering a change in medium, part of the wave is transmitted and part is reflected. In mathematical terms, when total internal reflection occurs, the reflection coefficient magnitude is unity. However, the reflected wave may experience a phase shift, which is represented as an exponential factor in the reflection coefficient equation:
\[\begin{equation}R = e^{-i2\phi}\end{equation}\]
where \(\phi\) is the phase shift that the reflected wave undergoes and \(i\) denotes the imaginary unit. Understanding the reflection coefficient enables us to predict the behavior of waves encountering different media, which has practical applications in designing optical devices.

Complex Angle

In certain scenarios, such as when dealing with total internal reflection, we encounter what are known as complex angles. This relates to the trigonometric functions reaching values greater than one, which is not possible for any real angle. In the context of our discussion, Snell's Law can predict a value of \(\sin\theta_{2}\) that exceeds one. This implies that \(\theta_{2}\) must be a complex angle. The imaginary part of this complex angle is significant because it gives rise to an important concept in optics known as the evanescent wave, which is a field that does not propagate but rather decays exponentially with distance from the boundary where total internal reflection occurs.

Phase Shift

Phase shift refers to the change in phase of a wave as it reflects off a boundary. This concept is essential in understanding the behavior of waves in various mediums, as the phase of a wave determines its position at any point in time. Total reflection is particularly interesting since the phase shift involved can result in an inverted or non-inverted reflected wave, depending on material properties and the incidence angle. The phase shift is apparent when considering the reflection coefficient as an exponential factor, where different angles of incidence will yield different phase shifts as seen in the problem we're examining.

Polarization of Waves

Polarization of waves is a property that describes the orientation of the oscillations in a wave, relative to the direction of propagation. In the context of electromagnetic waves, such as light, polarization refers to the direction in which the electric field vector oscillates. Total internal reflection can affect the polarization state of light waves. When light reflecting off a boundary does so at an angle greater than the critical angle, it may experience different phase shifts for different components of its electric field. This phenomenon can change the initial polarization of a wave, which explains why polarized sunglasses can reduce glare - they block light polarized in a certain direction.