Total constructive interference

Depending on the relative phase difference 8, total constructive interference, total destructive interference and any intermediate case between these two extrema are observed. Important special cases arise if the amplitudes of the two waves are equal, i.e. Ai = 1 21 and hence I = h = I- The total intensity at the interference location P can be summarized as follows ... 

When two or more waves pass through the same region of space, the phenomenon of interference is observed as an increase or a decrease in the total amplitude of the wave (recall Fig. 1.20). Constructive interference, an increase in the total amplitude of the wave, occurs when the peaks of one wave coincide with the peaks of another wave. If the waves are electromagnetic radiation, the increased amplitude corresponds to an increased intensity of the radiation. Destructive interference, a decrease in the total amplitude of the waves, occurs when the peaks of one wave coincide with the troughs of the other wave it results in a reduction in intensity. 

Under conditions of total internal reflection, guided modes are obtained due to the constructive interference between the light rays propagating through the waveguide30... 

The total path difference for the two waves scattered in the direction of kf by the two atoms is given by A = -j R (ki — kf) = jfRq. Constructive interference occurs for... 

FIGURE 3.8 X-rays diffracted off a crystal (viewed here as a regular array of atoms) will constructively interfere if multiple scattering paths reinforce. For example, in this picture, the two black atoms both produce scattered X-rays which would superimpose. The difference in the total path length (along the dashed paths) would be 2d sin 0. So the constraint for constructive interference is NX = 2s sin 0. 

The scattered X-rays mutually interfere, either destructively or constructively. Almost total destructive interference occurs in all but certain very specific directions in which constructive interference is possible (Sears et al., 1982). Constructive interference in a given direction results in a diffraction beam that is intense relative to the sum of all other rays scattered in the same direction. 

For large mode numbers A > 70 a simple plane wave model (PWM) has been deduced. Its main idea is, that the light wave circulates around exploiting the process of multiple total internal reflection (TIR, see Fig. la) and finally interferes with itself when having completed one full circulation within the resonator. To enforce constructive interference the total phase... 

If we do this, then the product of the transform of the object and the lattice becomes, as in Figure 5.4, simply the line of points, where each point serves as an identical source of a common wave corresponding to the scattering of the entire continuous object for some diffraction vector s. Although the lattice points produce a wave for any and all diffraction vectors s = (k — ko), because the waves arise from points in a lattice, the waves cancel, or sum to zero except when all the points belong to a family of planes hkl for which Bragg s law is satisfied, that is, when s = h. When this condition is met, the waves emitted from each point constructively interfere and sum in an arithmetic manner. The lattice then multiplies the resultant wave from the object, the atoms within the unit cell, by the total number of unit cells in the crystal and allows us to observe it, but only for specified values of s, namely only at those points in diffraction (Fourier transform) space where s = h. 

This is illustrated in Fig, 44 for a system similar to that of Fig. 4 except for the presence of a second totally symmetric mode. The solid line in Fig. 44 represents the fundamental of mode 1 for F = 0.2 (as in Fig. 4). Relative to the fundamental of Fig. 4, it shows a strong reduction of the V1V2 = 10 peak and the appearance of a 01 peak separated from the 10 peak by a sharp minimum due to interference. The 20 peak is enhanced, the 11 and 02 peaks are virtually absent, whereas the 30, 21, 12, and 03 peaks form one broad maximum, as a result of constructive interference. If F is increased to 0.5 (broken line), the REP shows simply three maxima of decreasing intensity, but their spacings are irregular and do not correspond to either coi or co2, nor do these maxima coincide with the absorption maxima. 

Figure 7.13 I Simple sine waves illustrate the concept of interference between waves. In the upper panel (a), the two identical sine waves on the left are totally in phase with one another, meaning that their peaks and valleys Une up. Adding these two waves will produce constructive interference, and the resulting wave will have twice the amplimde of the original waves. In the lower panel (b), the two waves are out of phase, so that the peaks of one correspond to the valleys of the other. Combining these waves will result in destructive interference, where the two waves effectively cancel one another and the result is a flat Une.

Because the electric and magnetic fields must be continuous across the interface, the radiation must penetrate a finite distance into the second medium even in the case of total internal reflection. Constructive interference of the incident and reflected fields in this situation creates a wave of evanescent ( vanishing ) radiation that propagates parallel to the interface but drops off quickly in amplitude beyond the interface. The fall-off of the field with distance (z) in the second medium is given by E = Eo jjp -zld) with... 

An important phenomenon that can occur with electromagnetic waves is interference. If several waves are present, the total electrical field at any place and time is the sum of the electrical fields of each of the waves. If the field vectors point in the same direction and are appropriately in phase, they add up to a stronger field, which is called constructive interference if they point in opposite directions, they cancel each other, which is referred to as destructive interference. If the phase relationship between the different waves does not change over time, constmctive and destructive interference will lead to areas of high and low intensity, respectively. In that case, the waves are said to be coherent. 

---