Linear Media

5 Key excerpts on "Linear Media"

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  eBook - ePub

  Light Propagation in Linear Optical Media

  • Glen D. Gillen, Katharina Gillen, Shekhar Guha(Authors)
  • 2017(Publication Date)
  • CRC Press

    (Publisher)

  2 Electromagnetic Waves in Linear Media

  In this chapter we present mathematical models describing the vector electric and magnetic fields for light propagating within a linear medium. Mathematical models presented cover:

  1. General forms for Maxwell's equations and the wave equation
  2. Maxwell's Equations for source- free media
  3. Maxwell's equations for vacuum
  4. Polarized light fields

  We then present and explain a few of the common forms of light that are solutions to these equations.

  To simplify the mathematics for this chapter, we place the following restrictions on the propagation of the light through the medium:

  • the volume through which the light is propagating is composed of only a single medium
  • the medium’s electric permittivity, ∈, and magnetic permeability, µ , are both constants
  • there are no boundary conditions for the volume, nor diffractive or spatially limiting elements within the volume

  2.1 Maxwell's Equations in Linear Media

  As discussed in Sections 1.2–1.4, when electromagnetic waves propagate through a medium other than a vacuum the electrodynamic properties of the medium must be taken into consideration. Here, we will assume that the medium can have any one, two, or all three of the following properties:

  1. the electric field of the EM wave, Ẽ , induces a dipole moment per unit volume,
     P ˜
     , within the medium,
  2. the magnetic field of the EM wave,
     H ,

     ˜
     induces a magnetic moment per unit volume,
     M ˜
     , within the medium, or
  3. the medium can have free charges per unit volume, or a free charge density, ρ .

  The vectors here are written with the tilde sign to signify their time dependence.

  The response of the medium to the incident electromagnetic fields can either be a linear response or a nonlinear response. In general, the magnitude of the induced polarization in the medium, P

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  eBook - ePub

  Materials and Acoustics Handbook

  • Michel Bruneau, Catherine Potel, Michel Bruneau, Catherine Potel(Authors)
  • 2013(Publication Date)
  • Wiley-ISTE

    (Publisher)

  Part 1 Homogenous and Homogenous Stratified Media: Linear Model of Propagation Chapter 1 Equations of Propagation 1 1.1. Introduction Acoustic (or elastic) waves are disturbances of a deformable medium (fluid or solid) propagating step by step into it by the actions of elementary particles on their neighbors. They do not exist in a vacuum. We will consider the medium as continuous (continuum mechanics), that is to say we will take an interest in elementary volumes which are much larger than atoms or molecules; it will be assumed that this volume contains a great number of these elements. We will sometimes name this volume “fluid or solid particles”. The acoustic equations will therefore result from those of continuum mechanics. We will first consider the medium as homogenous. In the absence of acoustic waves, it is in equilibrium. The acoustic waves will disturb this equilibrium but this disturbance will be small and will allow the equations to be linearized. The movement of the particles related to the wave will be described by a specific number of variables which will depend on the nature of the medium studied. 1.1.1. Fluid medium In the case of fluid media, the classic variables we are interested in are the vector of the particle speed, the p ressure ρ and the density ; consequently, there are three variables among which one is vectorial, and we need the same number of equations in order to solve the problem. We will assume a certain number of simplifying hypotheses; we will specifically neglect the viscosity and will suppose that, in the absence of acoustic waves, the fluid is at rest (no flow) and that there is no external force. Then the variables considered satisfy three equations

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  eBook - ePub

  Birefringent Thin Films And Polarizing Elements (2nd Edition)

  • Martin W McCall, Ian J Hodgkinson, Qihong Wu(Authors)
  • 2014(Publication Date)
  • ICP

    (Publisher)

  Spatial and temporal derivatives of these parameters are connected in a group of four equations known as Maxwell’s equations. In differential form Maxwell’s equations can be written as:

  Electric and magnetic fields provide the driving forces in optical media, and the consequences include electric current due to the motion of conduction electrons, polarization due to small relative displacements of bound charges, and magnetization due to induced magnetic moments. If an outcome such as electric current density is proportional to the strength of the driving force, J = σ E , then the medium is said to exhibit a linear response; alternatively the statement that Ohm’s law is obeyed would convey the same meaning in this example. The constant of proportionality in the relation used to link J and E is called the conductivity. In Table 2.1 we have listed the most important linear relationships for isotropic optical media and the corresponding proportionality constants. We will frequently be dealing with anisotropic dielectric media for which the relation D = ε 0 ε E in Table 2.1 must be replaced by the more general linear relation D = ε 0 ε E , where ε is the relative permittivity tensor. This book deals exclusively with linear optical materials [10 ]. Strictly speaking the relationships given in Table 2.1 are only correct when the fields are expressed in the frequency domain where, for example, D (ω ) = ϵ (ω )E (ω ), and medium dispersion is accounted for. In the temporal domain, account should be taken of the response time and history of the medium leading to the relationships being expressed as convolution integrals, e.g. However, if the medium responds more or less instantaneously, then the expressions in Table 2.1 are a good approximation in the time domain.

  2.2 Propagation in Free Space. Mathematical Methods

  In free space we have ρ = 0, P = 0, M = 0, J = 0, and Maxwell’s equations simplify to:

  Table 2.1 Linear relationships for isotropic optical media.

  Maxwell’s first and second equations, Eq. (2.2) and Eq. (2.3), are particularly significant for electromagnetic waves because they connect spatial changes in one field to temporal changes in the other. Six equations are implied, and the form is illustrated by writing out one pair; the others can be obtained by cyclic permutations of the subscripts x, y and z

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  eBook - ePub

  Engineering Optics with MATLAB®

  • Ting-Chung Poon, Taegeun Kim(Authors)
  • 2006(Publication Date)
  • WSPC

    (Publisher)

  Chapter 3

  Beam Propagation in Inhomogeneous Media

  3.1 Wave Propagation in a Linear Inhomogeneous Medium

  Thus far, as in Chapter 2 , we have only considered wave propagation in a homogeneous medium, characterized by a constant permittivity . In inhomogeneous materials, the permittivity can be a function of the spatial coordinates x, y and z, i.e., (x, y, z). To study wave propagation in inhomogeneous materials, we return to Maxwell's equations (2.1-1) -(2.1-4) and rederive the wave equation. Our starting point is Eq. (2.2-4) which we rewrite here for c = 0:

  Now, from Eq. (2.1-1) with ρv = 0, and Eq. (2.1-12a) , we have

  Substituting Eq. (3.1-2) into Eq. (3.1-1) yields

  The right-hand side of the above equation is in general non-zero when there is a gradient in the permittivity of the medium, such as the case in guided-wave optics. However, if the spatial variation of the refractive index is small over the distance of one optical wavelength, the term / ≈ 0 [Marcuse, 1982]. This approximation is of a similar nature as the paraxial approximation. If we were content with this approximation, we can study the propagation of light in inhomogeneous media by neglecting the right-hand side of Eq. (3.1-3) , giving us the following wave equation to solve:

  where = (x, y, z). Note that Eq. (3.1-4) is similar to the homogeneous wave equation for the electric field Eq. (2.2-10) derived earlier. For notational convenience, we return to our generic dependent variable (x, y, z, t) and adopt

  as our model equation, where we have assumed μ = μ0 for simplicity.

  3.2 Optical Propagation in Square-Law Media

  A square-law medium with an index of refraction of the form n2 (x, y) = – n2 (x2 + y2 ) has been studied in Chapter 1 using ray optics. Equivalently, we can incorporate the inhomogeniety through a square-law permittivity profile of the form [Haus (1984)]

  We wish to study the propagation of arbitrary beam profiles through the inhomogeneous medium modeled by Eq. (3.2-1) . However, analytical solutions of Eq. (3.1-5) with Eq. (3.2-1)

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  eBook - ePub

  Fundamentals of Liquid Crystal Devices

  • Deng-Ke Yang, Shin-Tson Wu(Authors)
  • 2014(Publication Date)
  • Wiley

    (Publisher)

  2 Propagation of Light in Anisotropic Optical Media 2.1 Electromagnetic Wave In wave theory, light is electromagnetic waves propagating in space [1–3]. There are four fundamental quantities in electromagnetic wave: electric field, electric displacement, magnetic field, and magnetic induction. These quantities are vectors. In the SI system, the unit of electric field is volt/meter ; the unit of electric displacement is coulomb/meter 2, which equals newton/volt · meter ; the unit of magnetic field is ampere/meter, which equals newton/volt · second, and the unit of magnetic induction is tesla, which equals volt · second/meter 2. In a medium, the electric displacement is related to the electric field by (2.1) where ε o = 8.85 × 10 −12 farad/meter = 8.85 × 10 −12 newton/volt 2 and is the (relative) dielectric tensor of the medium. The magnetic induction is related to the magnetic field by (2.2) where μ o = 4 π × 10 −7 henry/meter = 4 π × 10 −7 volt 2 · second 2 /newton · meter 2 is the permeability of vacuum and is the (relative) permeability tensor of the medium. Liquid crystals are non-magnetic media, and the permeability is close to 1 and approximately we have, where is the identity tensor. In a medium without free charge, the electromagnetic wave is governed by the Maxwell equations: (2.3) (2.4) (2.5) (2.6) When light propagates through more than one medium, at the boundary between two media, there are boundary conditions: (2.7) (2.8) (2.9) (2.10) At the boundary, the normal components of and and the tangential components of and are continuous. These boundary condition equations are derived from the Maxwell equations. Figure 2.1 Schematic diagram showing the electromagnetic fields at the boundary between two media; represents,,, and ; is a unit vector along the normal direction of the interface; n: normal component and t: tangential component. We first consider light propagating in an isotropic uniform medium where and

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