Monochromatic waves

Monochromatic Waves (1.14) A monochromatic e.m. wave Vcj r,t) can be decomposed into the product of a time-independent, complex-valued term Ucj r) and a purely time-dependent complex factor expjojt with unity magnitude. The time-independent term is a solution of the Helmholtz equation. Sets of base functions which are solutions of the Helmholtz equation are plane waves (constant wave vector k and spherical waves whose amplitude varies with the inverse of the distance of their centers. 

According to De Broglie an electron in a Bohr orbit is associated with a standing wave. To avoid self destruction by interference an integral number of wavelengts are required to span the orbit of radius r, which implies n — 2nr, or nh/2n = pr, which is the Bohr condition. As a physical argument the wave conjecture is less plausible, but not indefensible. One possible interpretation considers the superposition of several waves rather than a single monochromatic wave to simulate the behaviour of a particle. 

First, we rewrite expression (5.14), which is valid for plane monochromatic waves, in a more general way, valid for a more general wavefront, as follows ... 

Figure 2.7 Si 220 reflection with a plane incident monochromatic wave, (a) duMond diagram showing the angle at which the wave will diffract, and a 2° angular aperture that easily allows it to pass, (b) the corresponding real-space geometry...

In the case of a common upper level the fluorescence from this level due to excitation by the monochromatic wave will reflect this selective population with molecules having only velocity components inside the range Auz. Thus, observing the fluorescence in z direction on any transition from this upper level yields linewidths much smaller than the normal doppler width 322). In this way lines which are not resolved in normal fluorescence spectroscopy can be separated even if their frequency difference is much less than their doppler width 323). 

Consider a plane monochromatic wave with angular frequency and wave number k which is propagating in the z direction in a nonabsorbing medium. In discussions of polarization it is customary to focus attention on the electric field E ... 

Equation (2.78) describes an ellipse, the vibration ellipse (Fig. 2.11). If A = 0 (or B = 0), the vibration ellipse is just a straight line, and the wave is said to be linearly polarized , the vector B then specifies the direction of vibration. (The term plane polarized is also used, but it has become less fashionable in recent years.)If A = B and A B = 0, the vibration ellipse is a circle, and the wave is said to be circularly polarized. In general, a monochromatic wave of the form (2.77) is elliptically polarized. 

Although the ellipsometric parameters completely specify a monochromatic wave of given frequency and are readily visualized, they are not particularly conducive to understanding the transformations of polarized light. Moreover, they are difficult to measure directly (with the exception of irradiance, which can easily be measured with a suitable detector) and are not adaptable to a... 

Although a strictly monochromatic wave, one for which the time dependence is exp( — itot), has a well-defined vibration ellipse, not all waves do. Let us consider a nearly monochromatic, or quasi-monochromatic beam ... 

Our fundamental task is to construct solutions to the Maxwell equations (3.1)—(3.4), both inside and outside the particle, which satisfy (3.7) at the boundary between particle and surrounding medium. If the incident electromagnetic field is arbitrary, subject to the restriction that it can be Fourier analyzed into a superposition of plane monochromatic waves (Section 2.4), the solution to the problem of interaction of such a field with a particle can be obtained in principle by superposing fundamental solutions. That this is possible is a consequence of the linearity of the Maxwell equations and the boundary conditions. That is, if Ea and Efc are solutions to the field equations,... 

In order to obtain an insight into the intramolecular energy losses of photo-ejected electrons, their kinetic energy distributions have been measured by the method of the retarding field in a cylindrical condenser, schematically shown in Figure 8, for a sequence of monochromatic wave-... 

Exercise. The equation for a monochromatic wave in a three-dimensional medium with random refractive indexis... 

These solutions can be divided into four independent monochromatic waves ... 

For a monochromatic wave of frequency oa in an inhomogeneous and isotropic medium of refraction index n r), the electromagnetic field satisfies Eq. (1) with k given by... 

This equation fixes the optical frequency u)n = 2Tmvp(u)n)/2L and the wave number k(u)n) = u>n/vp(u)n) of the nth cavity mode, where vp(u>n) is the phase velocity for a monochromatic wave at u)n. The following expansion about some mean frequency u>m is generally used to take dispersion into account ... 

Oscillations with only one frequency are monochromatic waves. Thus each normal mode of oscillation [each term in equation (8)] defines a monochromatic wave. There are special shapes of chambers for which more than one mode may have the same frequency this is called degeneracy and admits an infinite variety of monochromatic wave forms (for example, tangential modes in cylindrical chambers). Most of the normal modes describe standing waves, waves having nodal points for the velocity (points where the velocity is always zero) and for the amplitude of the pressure oscillations. Thus, according to equation (8), longitudinal modes have pressure nodes at nz/l = 1, I,..., and they have velocity nodes at nz/l = 0, 1, 2,..., as... 

A clearer interpretation of the boundary work may be obtained by neglecting convection and homogeneous dissipation and by focusing attention on a monochromatic wave field of frequency co. A boundary at which a rigid-wall condition (v n = 0) or an isobaric condition (p = 0) is exactly applicable clearly has

n = 0 and therefore no boundary work. For... 

The assumptions of the kinematical theory - that the incident wave is monochromatic and plane and that there is no absorption of either the transmitted wave or the scattered waves - are reasonable assumptions with which to begin a theory. However, the other assumptions - that a scattered wave is never rescattered and that there is no interaction between the transmitted and scattered waves - are gross oversimplifications of the physical situation. For example, consider a plane monochromatic wave incident upon a crystal plate at such an angle that the Bragg law is satisfied for a set of planes approximately normal to the crystal plate, as shown in Figure 3.20. It is clear from this diagram that at A the diffracted wave Si in the crystal is rediffracted so that it travels in the same direction as the transmitted wave T. This rediffracted wave is denoted S2. There is no reason why S2 should not be also rediffracted at B to produce the wave S3, as shown. Thus, the assumption that the diffracted wave, once produced in the crystal, is never rediffracted, is clearly unacceptable. 

By assuming harmonic forces and periodic boundary conditions, we can obtain a normal mode distribution function of the nuclear displacements at absolute zero temperature (under normal circumstances). The problem is then reduced to a classic system of coupled oscillators. The displacements of the coupled nuclei are the resultants of a series of monochromatic waves (the normal modes). The number of normal vibrational modes is determined by the number of degrees of freedom of the system (i.e. 3N, where N is the number of nuclei). Under these conditions the one-phonon dispersion relation can be evaluated and the DOS is obtained. Hence, the measured scattering intensities of equations (10) and (11) can be reconstructed. 

Let us consider an isolated molecule perturbed by an electromagnetic field. According to the semiclassical approach, the external radiation is described as a plane monochromatic wave traveling with velocity c and obeying the Maxwell equations [21] (i.e., the fields are not quantized). 

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