Full wave vector maxwell equation modeling of the selffocusing of ultrashort optical pulses in a nonlinear kerr medium exhibiting a finite response time richard w ziolkowski and justin b. Optical fiber a single solid dielectric of two concentric layers. The nonplane wave solutions that we are looking for are of the form u 0 ux. There is a simple set of complex traveling wave solutions to this equation. This is followed by the introduction of the stokes parameters and a discussion of their ellipsometric content. The wave equation derive the wave equation that a disturbance propagates without changing it shape. Derivation of the wave equation from maxwells equations.
In this case the wave number becomes a vector, k, and the harmonic solution is. The theory is applicable to apertures larger than a wavelength situated in dark, perfectly absorbing screens and for points of observation in the right halfspace at distances greater than a wavelength beyond the aperture. Vector fields, vector derivatives and the 3d wave equation. Aeikxt the harmonic phase written in cartesian coordinates, is required to be.
The optical properties of solids provide an important tool for studying energy band structure, impurity levels, excitons, localized defects, lattice vibrations, and certain magnetic. Feb 15, 2016 lectures on theory of microwave and optical waveguides weng cho chew 1 fall 2015 1updated february 15, 2016. More on maxwell wave equations boundary conditions poynting. Observation and differentiation of unique highq optical resonances near zero wave vector in macroscopic photonic crystal slabs jeongwon lee, bo zhen, songliang chua, wenjun qiu, john d. Fullwave vector maxwell equation modeling of the self.
The wave at different times, once at t0, and again at some later time t. Existence of solutions to general nonconstant coe cient linear wave equation 18 6. Thus, the number of photons per unit volume must be proportional to e2, and the proportionality constant must depend on the frequency of the wave. To measure sound level or magnitude one typically uses a decibel db 110 bel. We can think of a wave front as a line along the crest of the wave. Waterwaves 5 wavetype cause period velocity sound sealife,ships 10. The wave equation the function fz,t depends on them only in the very special combination zvt. In this case t x u xy 2 2 where xy c44 xy and x u xy wave equation is 2 2 44 2 2 x c u t u x v c44 t. The above equation implies that when a wave gets refracted into a denser medium v 1 v 2 the wavelength and the speed of propagation decrease but the frequency f v.
We start by formulating the macroscopic maxwell equations and constitutive relations and discussing the fundamental timeharmonic plane wave solution that underlies the basic optical idea of a monochromatic parallel beam of light. Modes are referred to as the characteristic waves of the structures because their field vector satisfies the homogenous wave equation in all the media that make up the figure 1. It is customary in the theory of optical fibers to make the weakly guiding approximation n. Before getting into maxwells equations and the wave equation for light, lets do a warmup. For this case the right hand sides of the wave equations are zero. The change to vector notation produced a symmetric mathematical representation. The wave vector is related to the angular frequency by. The electromagnetic wave from maxwells equations contd 2 2 t e e w w u u 2 2 2 t e e e o o w w x ph xe 0 using the vector identity becomes, in free space and we are left with the wave equation 0 2 2 2 w w t e e p oh o. Let light pass through the plate with the wave vector perpendicular to the. Do note that sound as seen next is a perturbation of pressure in time and space in a stationary calm medium.
Wave propagation in cylindrical optical waveguides waveguides for signal transmission must propagate a wave longitudinally in the zdirection. Orthogonal polarization, the same sample also exhibits a negative refractive index in the pdfgrabber 3 visible the phase space. This explicitly does depends on the frequency of the wave. These new equations only have terms which contain therefractiveindexn2 and e or h,thus. These two formulations always give the same answer. Linear wave equation via fundamental solution and representation formula 7 3. Both components of outgoing scattered wave are plane waves with wavevector k energy conservation. Chapter 2 maxwellbloch equations mit opencourseware. When the elasticity k is constant, this reduces to usual two term wave equation u tt c2u xx where the velocity c p k.
When one recalls that the characteristic frequencies for the fluctuations in the dielectric polarizability are quite small compared with the optical frequency 46. The definition of the difference of two vectors is evident from the equation for the. Judkins department of electrical and computer engineering, university of arizona, tuscon, arizona 85721. In the total internal reflection the incident wave penetrates the less dense medium to certain depth of the order of wavelength which causes phase shift between the incident and the reflected wave 1. In general, waves propagating in birefringentcrystal change polarization. Wave equation in crystals with respect to the principal axes, dispersion relation for planewaves and field eigenpolarizations, wavevector or normal surface, intersecting shells and. Now let us derive a simplified version of the vector wave equation.
The observables are represented by hermitian operators a. Now, we could find a similar firstorder differential equation for e 0, and then solve the two coupled equations. The state of a system is described by a state vector. To measure sound level or magnitude one typically uses a.
Chapter 6 maxwells equations for electromagnetic waves. Optical fiber wave guiding to understand transmission mechanisms of optical fibers with dimensions approximating to those of a human hair. A vector wave equation for the electric field note the vector symbol over the e. Each normal coordinate qk represents an independent vibrational mode of the lattice with wavenumber k, which is known as a normal mode. The homogeneous form of the equation, written in terms of either the electric field e or the magnetic field b, takes the form. Matrix description of wave propagation and polarization. A more general solution is left to the next section. The electromagnetic wave equation is a secondorder partial differential equation that describes the propagation of electromagnetic waves through a medium or in a vacuum. Simple derivation of electromagnetic waves from maxwells equations. Refraction of a plane wave by a thin prism we consider a plane wave passing through a thin prism. Publication of this book was made possible in part by the virginia tech university libraries open education initiative faculty grant program.
Fiber acts as an open optical waveguide may be analyzed using simple ray theory geometric optics. This is really three independent wave equations, one each for the x, y, and zcomponents of e. The electromagnetic spectrum is a plot of the field magnitudes or energies as a function of wavelength. Now develop the general one d wave e xx t x x yyx fx f x xx xx x yy yx fxf xxx xxxxx quation x. Observation and differentiation of unique highq optical. Both vector fields shown in the examples of divergence have zero curl, since. Consistent with geometrical optics, its magnitude is constrained to be proportional to the refractive index n 2. Relativistic wave equations and their derivation 5. Returning to fundamental solutions to the linear wave equation 9 3. The origins and frequency dependency of the dielectric.
Wave equation in crystals with respect to the principal axes, dispersion relation for planewaves and field eigenpolarizations, wavevector or normal surface, intersecting shells and the optic axes, two field eigenpolarizations for. Wave equations hold for each component of the field vector e or h, every component satisfying the scalar wave equation. The interface is located at z 0and the incident wavevector k0,there. Chapter maxwells equations and electromagnetic waves. The wave vector k can have an arbitrary orientation and is related to the frequency as k. The two forms are mathematically equivalent and useful in. Preface this textbook treats some of the necessary prerequisites for the analysis of spherical vector wave solutions to the maxwell equations. This section develops the basic wave equations used in radio and optics. Thus any cartesian component of e or b obeys a classical wave equation of the form.
Maxwells equations before delving into the actual wave equations. We shall therefore proceed by constructing a wave equation for spin1 2 fermions and only return to the kleingordon equation in connection with motion in a coulomb potential mesons. S is the poynting vector and indicates the direction and magnitude of power flow in the em field. In electromagnetic wave theory, the conventional maxwells equations are formulated via vector notation in two forms of integral equations and differential forms. A standing electromagnetic wavedoes not propagate, but instead the electric and magnetic fields execute simple harmonic motion perpendicular to the would be direction of propagation.
This equation is mathematically the same as the harmonic oscillator equation and has solutions where is a constant vector. These equations do not impose any conditions on the medium and are therefore always valid. Solving the wave equation in second order but, instead of doing that, lets see if we can gain some physical insight by making another simplifying assumption. The wave equation handbook of optical systems wiley. Now let us derive a simplified version of the vector wave equa. Origins of the electric permittivity and magnetic permeability.
Necessary to consider the optical waveguiding of a cylindrical glass fiber. The electromagnetic wave from maxwells equations contd 2 2 t e e w w u u 2 2 2 t e e e o o w w x ph xe 0 using the vector identity becomes, in free space and we are left with the wave equation. New approach to optical diffraction tomography yielding a. It has been shown last week that for any vector a where is the laplacian operator thus. Maxwells four equations into two uncoupled wave equations. Inner layer core of radius aand refractive index n 1. Solids with more than one atom in the smallest unit cell exhibit two types of phonons. Last lecture stops, pupils, windows optical magnifiers microscopes telescopes this lecture equations of traveling waves harmonic waves plane waves em wave equations. Doublenegative at 8 nm and singlenegative at 772 nm. When that is true, the function fz,t represents a wave of fixed shape traveling in the z direction at speed v. It continues with a few specic simple solutions such as the inniteextent plane wave.
It is an excerpt of the much more detailed textbook by the author, viz. There is more information contained in maxwells equations than there is in the wave equation. We will derive the wave equation from maxwells equations in free space. Chapter 1 optical properties of plasmonic materials. It is a threedimensional form of the wave equation.
More on maxwell wave equations boundary conditions. Vector wave propagation and diffraction formulated in this way are entirely consistent with huygenss principle. Approximative solutions to the wave equation geometrical optics and the eikonal equation. Outer layer cladding refractive index n 2 n 2 optical fiber is a long cylindrical dielectric waveguide, usually. A solution of the wave equation longitudinal plane wave ux,t aeiqx t where q wave vector. Which can be interpreted as the sum of 3n independent oscillator hamiltonians, one for each wave vector and polarization. Solid state physics part ii optical properties of solids.
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