Plane harmonic wave

Up to now we have considered only harmonic generation by a plane wave. Harmonic generation by a focused laser beam was treated by Kleinman et al (1966), Ward and New (1969) and applied for THG in liquids and in air (environmental effects) by Meredith et al (1983b), Kajzar and Messier... 

We use s, p, and d partial waves, 16 energy points on a semi circular contour, 135 special k-points in the l/12th section of the 2D Brillouin zone and 13 plane waves for the inter-layer scattering. The atomic wave functions were determined from the scalar relativistic Schrodinger equation, as described by D. D. Koelling and B. N. Harmon in J. Phys. C 10, 3107 (1977). 

The moving harmonic wave t) in equation (1.3) is also known as a plane wave. The quantity kx — CDt) is called the phase. The velocity (o/k is, known as the phase veloeity and henceforth is designated by Uph, so that... 

Equation (1.8) represents a plane wave exp[i(A x — mt)] with wave number k, angular frequency m, and phase velocity m/A, but with its amplitude modulated by the function 2 cos[(AA x — Amt)/2]. The real part of the wave (1.8) at some fixed time to is shown in Figure 1.2(a). The solid curve is the plane wave with wavelength X = In jk and the dashed curve shows the profile of the amplitude of the plane wave. The profile is also a harmonic wave with wavelength... 

Expressions for the electric and magnetic fields can likewise be obtained. These plane-wave solutions are then expanded in terms of spherical harmonics... 

Fourier transformation of the spherical harmonic functions is accomplished by expanding the plane wave exp(27r/ST) in terms of products of the spherical harmonic functions. In terms of the complex spherical harmonics Ylm 6, [Pg.68]

The delta function corresponds to Einstein s equation, which says that the kinetic energy of the emitted electron Ef equals the difference of the photon energy h(a and the energy level of the initial state of the sample, The final state is a plane wave with wave vector k, which represents the electrons emitted in the direction of k. Apparently, the dependence of the matrix element 1 j) on the direction of the exit electron, k, contains information about the angular distribution of the initial state on the sample. For semiconductors and d band metals, the surface states are linear combinations of atomic orbitals. By expressing the atomic orbital in terms of spherical harmonics (Appendix A),... 

In addition to irradiance and frequency, a monochromatic (i.e., time-harmonic) electromagnetic wave has a property called its state of polarization, a property that was briefly touched on in Section 2.7, where it was shown that the reflectance of obliquely incident light depends on the polarization of the electric field. In fact, polarization would be an uninteresting property were it not for the fact that two waves with identical frequency and irradiance, but different polarization, can behave quite differently. Before we leave the subject of plane waves it is desirable to present polarization in a systematic way, which will prove to be useful when we discuss the polarization of scattered light. 

EXPANSION OF A PLANE WAVE IN VECTOR SPHERICAL HARMONICS... 

Expansion of a plane wave in vector spherical harmonics is a lengthy, although straightforward, procedure. In this section we outline how one goes about determining the coefficients in such an expansion. 

The desired expansion of a plane wave in spherical harmonics... 

In Chapter 4 a plane wave incident on a sphere was expanded in an infinite series of vector spherical harmonics as were the scattered and internal fields. Such expansions, however, are possible for arbitrary particles and incident fields. It is the scattered field that is of primary interest because from it various observable quantities can be obtained. Linearity of the Maxwell equations and the boundary conditions (3.7) implies that the coefficients of the scattered field are linearly related to those of the incident field. The linear transformation connecting these two sets of coefficients is called the T (for transition) matrix. I f the particle is spherical, then the T matrix is diagonal. 

Since, in this causal model, the extended wave 0 represents a real physical finite wave with well-defined energy, it seems natural to represent it by a suitable mathematical form. At the time when de Broglie put forth his causal interpretation of quantum mechanics, it was necessary for him to construct a finite wave using the Fourier analysis, namely, the multiplicity of harmonic plane waves, infinite in space and time, summing up and giving origin to a wavepacket. 

It seems quite natural to describe the extended part of a quantum particle not by wavepackets composed of infinite harmonic plane waves but instead by finite waves of a well-defined frequency. To a person used to the Fourier analysis, this assumption—that it is possible to have a finite wave with a well-defined frequency—may seem absurd. We are so familiar with the Fourier analysis that when we think about a finite pulse, we immediately try to decompose, to analyze it into the so-called pure frequencies of the harmonic plane waves. Still, in nature no one has ever seen a device able to produce harmonic plane waves. Indeed, this concept would imply real physical devices existing forever with no beginning or end. In this case it would be necessary to have a perfect circle with an endless constant motion whose projection of a point on the centered axis gives origin to the sine or cosine harmonic function. This would mean that we should return to the Ptolemaic model for the Havens, where the heavenly bodies localized on the perfect crystal balls turning in constant circular motion existed from continuously playing the eternal and ethereal harmonic music of the spheres. 

These harmonic plane waves, which supposedly existed over the full spectrum of time, whether past, present, or future, and also in the whole gamut of infinite space, have, of course, no real physical existence. They are mere abstract entities, existing only in our minds. It would be necessary to dispose of the whole infinity of space and time in order to produce them. This logical requirement certainly appears impossible to achieve as a goal since practical real physical devices are always finite in space and time. Now, the time is ripe for improving our description of nature with more appropriate tools. 

It is well known that any reasonable function can be Fourier-represented as a sum of infinite, in space and time, harmonic plane waves (i.e, sinus and cosinus). The more localized the function representing the particle, the more waves the needed to reconstruct it. In the limiting case, when the particle is precisely localized, Ax = 0, corresponding to a Dirac delta function, the number of waves necessary to build it up reaches infinite values. Since each wave is associated with one velocity, this means that a precisely localized particle has an associated infinitude of velocities, that is, an infinite error for the velocity Av = oo. If, instead of a well-defined position, one wishes to have a particle with a precise velocity Av = 0, only one single wave is to be used. Since the harmonic wave with a well-defined velocity is infinite in either space or time, this means that the particle is somehow spread over all space, implying that is its position is completely unknown, Ax = oo. 

The usual uncertainty relations are a direct mathematical consequence of the nonlocal Fourier analysis therefore, because of this fact, they have necessarily nonlocal physical nature. In this picture, in order to have a particle with a well-defined velocity, it is necessary that the particle somehow occupy equally all space and time, meaning that the particle is potentially everywhere without beginning nor end. If, on the contrary, the particle is perfectly localized, all infinite harmonic plane waves interfere in such way that the interference is constructive in only one single region that is mathematically represented by a Dirac delta function. This implies that it is necessary to use all waves with velocities varying from minus infinity to plus infinity. Therefore it follows that a well-localized particle has all possible velocities. 

The third contribution to the chemical potential is due to strain. If A and B atoms (ions) have different size, clustering results in elastic lattice distortions. By making a Fourier transformation, one can decompose the concentration profile into harmonic plane waves [D. DeFontaine (1975)]. The elastic energy contributions of these concentration waves are additive in the Unear elastic regime and yield Ea. Therefore, we may write... 

On the U(l) level, the plane wave is subjected to a multipole expansion in terms of the vector spherical harmonics, in which only two physically significant values of M in Eq. (761) are assumed to exist, corresponding to M = +1 and — 1, which translates into our notation as follows ... 

A complete set of standard time-harmonic solutions to Maxwell s equations usually involve the plane wave decomposition of the field into transverse electric... 

Kajzar and Messier have analyzed the THG from their cell described above. A brief overview of their analysis is given here. The cell is comprised of two thick wedge windows and a thin liquid wedge compartment. Since the windows are thick, they are considered to be infinite nonlinear media. Since the liquid chamber is thin, the laser field is treated as a plane wave in that region. The third harmonic field at the output of the cell is the resultant of the fields generated in the three media... 

Equation (11) describes the time dependence of the displacements. In the harmonic approximation the displacements are plane waves... 

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