Surface of constant phase

We can talk about the wave-front associated with this wave as being a surface of constant phase traveling at the speed of the wave. This surface is related to our previous idea of the rays in that it represents the joined perpendiculars from a series of rays emitted from a single point. A perfectly spherical wave will converge to (or diverge from) a single point in space. 

Figure 8.7 Schematic diagram of the surfaces of constant phase in a wedge-shaped crystal...

E0exp( — k" x) and H0exp( —k" x) are the amplitudes of the electric and magnetic waves, and = k x — ut is the phase of the waves. An equation of the form K x = constant, where K is any real vector, defines a plane surface the normal to which is K. Therefore, k is perpendicular to the surfaces of constant phase, and k" is perpendicular to the surfaces of constant amplitude. If k and k" are parallel, which includes the case k" = 0, these surfaces coincide and the waves are said to be homogeneous if k and k" are not parallel, the waves are said to be inhomogeneous. For example, waves propagating in a vacuum are homogeneous. 

Let us briefly consider propagation of surfaces of constant phase. Choose an arbitrary origin O and a plane surface over which the phase is constant (Fig. 2.2). At time t the distance from the origin O to the plane is z, where k x = k z and k z — ut = . In a time interval At the surface of constant phase will have moved a distance A z, where... 

Thus, the velocity of propagation of surfaces of constant phase, the phase velocity v, is... 

The surfaces of constant phase, or wave fronts, of the scattered wave (8.33), the points on which satisfy... 

A plane wave exists at a given time, when all the surfaces of constant phase form a set of planes, each generally perpendicular to the propagation direction. Under these conditions Eq. (1.3) becomes Eq. (1.4), defining the unit vector / in the direction perpendicular to the wave plane, that is, in the direction of the wave propagation. 

The vector Rr is normal to the surfaces of constant phase, whereas kj is normal to the surfaces of constant amplitude. Indeed, a plane surface normal to a real vector K is described by r K = constant, where r is the radius veetor drawn from the origin of the reference frame to any point in the plane (see Fig. 

Equations (12.4)-(12.9) describe an outgoing transverse spherical wave propagating radially with the phase velocity v = cojk and having mutually perpendicular complex electric and magnetic field vectors. The wave is homogeneous in that the real and imaginary parts of the complex wave vector kx are parallel. The surfaces of constant phase coincide with the surfaces of constant amplitude and are spherical. Obviously,... 

Figure 5. Surfaces of constant phase and directions of vectors E (sum of the incident and scattered waves) in the close vicinity of a spherical particle. The incident wave propagates along the Z-axis (indicated by the wave vector k) and is polarized along the X-axis. The particle size parameter is x =4 and the refractive index is fh = 1.32 + 0.05 .

Direct calculations using the Lorentz Mie theory for spherical particles show that the surfaces of constant phase of the total field are funnel-shaped in the vicinity of the particle (Fig. 5). Such near-field properties are typical for spherical particle with other size and refractive index [26,28]. 

The fundamental ideas of interferometry are treated, at various levels of mathematical rigor, in standard optical texts (Hecht 2003 Bom et al. 1999). Conceptually, interferometry is most easily understood in terms of the wave nature of light, an electromagnetic wave (following Maxwell s equations) with a phase that changes by 2n radians every wavelength. A wavefront is a surface of constant phase (e.g., the peak, valley, or any other phase in Fig. 1). 

The equation cot — kr = const describes a plane normal to the wave vector k. This plane is characterized by a constant phase A = cot — const and is called the surface of constant phase or the wavefront. The wavefront travels in a medium in the direction k with the velocity... 

Planewave Wave configuration in which surfaces of constant phase form parallel flat planes. All the light in a planewave is travelling the same direction, perpendicular to the surface of the planes. 

Wavefront Surface of constant phase in a propagating wave. 

It can be shown [5.1,5.24] that in nonfocal resonators with large Fresnel numbers N the field distribution of the fundamental mode can also be described by the Gaussian profile (5.32). The confocal resonator with d = R can be replaced by other mirror configurations without changing the field configurations if the radius Rf of each mirror at the position zo equals the radius R of the wavefront in (5.37) at this position. This means that any two surfaces of constant phase can be replaced by reflectors, which have the same radius of curvature as the wave front - in the approximation outlined above. 

This section is devoted to a reconsideration of system (3.2.2), to obtain the frequency change as a power series in e. A slightly careful examination of Method I reveals that a major obstacle to systematic perturbation expansions lies in the fact that the surfaces of constant phase are generally curved in state space. Since the definition of the surfaces of constant phase is entirely at our disposal. 

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