Fiber Optic Tutorials

This is a continuation from the previous tutorial - Phase Velocity, Group Velocity, and Dispersion. As discussed in the tutorial of optical fields and Maxwell's equations, and linear optical susceptibility, dispersion in the susceptibility of a medium is caused by the fact that the response of the medium to excitation by an optical field does not decay instantaneously. The general characteristics of the medium can be understood from its impulse response. In general, the impulse response of a medium decays exponentially while oscillating at some resonance frequencies. There may exist several exponential relaxation constants and several oscillation frequencies for a given...

This is a continuation from the previous tutorial - Reflection and Refraction. For a monochromatic plane optical wave traveling in the z direction, the electric field can be written as \[\tag{159}\mathbf{E}=\pmb{\mathcal{E}}\exp(\text{i}kz-\text{i}\omega{t})\] where \(\pmb{\mathcal{E}}\) is a constant vector independent of space and time. This represents a sinusoidal wave whose phase varies with z and t as \[\tag{160}φ=kz-\omega{t}\] For a point of constant phase on the space- and time-varying field, φ = constant and thus kdz - ωdt = 0. If we track this point of constant phase, we find that it is moving with a velocity of \[\tag{161}\nu_p=\frac{\text{d}z}{\text{d}t}=\frac{\omega}{k}\] This is called the phase velocity of the...

This is a continuation from the previous tutorial - Gaussian Beam. The characteristics of reflection and refraction of an optical wave at the interface of two different media depend on the properties of the media. We first consider the simple case of reflection and refraction at the planar interface of two dielectric media that are linear, lossless, and isotropic. In this situation, the permittivities ε1 and ε2 of the two media are constant real scalars, while the permeabilities are simply equal to μ0 at optical frequencies. We assume that the optical wave is incident from medium 1 with a wavevector ki, while the reflected...

This part continues from the Propagation in an Anisotropic Medium tutorial. Because the wave equation governs optical propagation, the transverse field distribution pattern and its variation along the longitudinal propagation direction have to satisfy this equation in order for the wave to exist and to propagate. A well-defined field pattern that can remain unchanged as the wave propagates is called a mode of wave propagation. Such a transverse field pattern is known as a transverse mode. The optical modes that exist in a given medium are determined by the optical properties of the medium together with any boundary conditions imposed...

This part continues from the Propagation in an Isotropic Medium tutorial. In an anisotropic medium, the tensors χ and ε do not reduce to scalars. Therefore, \(\mathbf{P}\not\parallel\mathbf{E}\text{ and }\mathbf{D}\not\parallel\mathbf{E}\). As a result, (81) [from the propagation in an isotropic medium tutorial] is not true any more, and, in general, \[\tag{104}\boldsymbol{\nabla}\cdot\mathbf{E}\ne0\] Consequently, (82) cannot be used for propagation of a monochromatic wave in an anisotropic medium. Instead, (80) has to be used. \[\tag{80}\boldsymbol{\nabla}\times\boldsymbol{\nabla}\times\mathbf{E}+\mu_0\epsilon(\omega)\cdot\frac{\partial^2\mathbf{E}}{\partial t^2}=0\]   Anisotropic χ and ε In a linear anisotropic medium, both χ and ε are second-rank tensors. They can be expressed in the following matrix forms: \[\tag{105}\pmb{\chi}=\begin{bmatrix}\chi_{11}&\chi_{12}&\chi_{13}\\\chi_{21}&\chi_{22}&\chi_{23}\\\chi_{31}&\chi_{32}&\chi_{33}\\\end{bmatrix}\] and \[\tag{106}\pmb{\epsilon}=\begin{bmatrix}\epsilon_{11}&\epsilon_{12}&\epsilon_{13}\\\epsilon_{21}&\epsilon_{22}&\epsilon_{23}\\\epsilon_{31}&\epsilon_{32}&\epsilon_{33}\\\end{bmatrix}\] The relationships \(\mathbf{P}=\epsilon_0\pmb{\chi}\cdot\mathbf{E}\) and \(\mathbf{D}=\pmb{\epsilon}\cdot\mathbf{E}\) are carried out...