Monochromatic waves superposition

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. 

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,... 

A monochromatic wave is superposition of infinite number of plane waves one-or two-waves approximations can only be used for waves with k nil (far from the gaps). 

By definition the monochromatic wave is boundless in space. The real wave is always limited in space and is emitted during a limited interval of time, which is why it cannot be monochromatic in full measure. However, any real wave can be considered as a result of the superposition of a large number of strictly monochromatic flat waves. As a result of interference, in one part of space these waves strengthen each other, and in other parts extingnish. Such waves have some features that can be discovered using a simple model of superposition of two plane monochromatic waves. 

Let two plane cross-sectional polarized monochromatic waves with equal amplitudes be distributed along an axis x. Such waves are described by equations = A cos(ft)iZ - kiX) and 1 2 = A cos(cOiZ k2X). Because of the superposition principle a combined wave can be represented as ( = if 1 + ( 2 cos(coj - k x) + A cos(cOit - k-pc), or... 

We can see that the superposition of two monochromatic waves with equal amplitudes with slightly different frequencies and wavenumbers produces a new wave with variable in space and time amplitude... 

Nonmonochromatic Waves (1.16) Diffraction theory is readily expandable to non-monochromatic light. A formulation of the Kirchhoff-Fresnel integral which applies to quasi-monochromatic conditions involves the superposition of retarded field amplitudes. 

This observation may be explained as follows. A wave emitted from a point source with the spectral width Aco can be regarded as a superposition of many quasi-monochromatic components with frequencies within the interval Act). The superposition results in wave trains of finite length A c = =... 

A general pulse of coherent polychromatic electromagnetic radiation can be described as superposition of monochromatic plane waves. The vector potential and fields of such a pulse are then given as... 

The integration in Equation [4] considers the existence of several frequency combinations matching the condition = Ey cOy, within the bandwidth of the applied field. A single laser whose bandwidth is large enough to include frequency components that match 0) -0) = O) can drive the nonlinear response. However, more frequently the laser bandwidth is smaller than o) and different beams with distinct centre frequencies at needed to match the Raman resonance. Let us now consider a field composed of a superposition of quasi-monochromatic beams, i.e. the amplitude at a carrier frequency is modulated by a complex envelope that defines its spectral shape. The central frequencies are chosen to match the Raman resonant four-wave mixing scheme (see Table 1), which can be later adapted to other colour choices. Three laser beams are present of frequencies o)q, cOp, cOg, with 0) -0) = 0) as the only resonant frequency combination. 

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