Waves

Diffraction experiment with waves

In general it can be written the following expression for a wave:

f = Ae−iwt+ikx w=2pn: frequency of a wave k = 2π/l = p/h : wave vector

Intensity of a wave is given by the following quantity square: I = |φ|2. The general rule for calculation of complex number c square is as follows |c|2 = c·c* ; where c* is complex conjugate to complex number c; for instance complex conjugate to i is -i ( i ® -i ).

When substituting the first equation into expression for intensity one can write down: I = |φ|2 = |Ae−i(wt−kx). A*e+i(wt−kx)| = |A|2

There are two waves coming from two identical slits (as it's shown on illustration) and both waves layer in detector D giving total intensity I. The intensity of two overlaped waves φ1 and φ2 having the same frequency but different phases is as follows Phase ist dann:

φ1 = A1e−iwt+ikx and f2 = A2e−iwt+ikx+ij where j = phase difference

I = |φ1+φ2|2 = (φ1+φ2) · (φ1*+φ2*) = |φ1|2 + |φ2|2 + φ1*·φ2 + φ1·φ2*

I = I1 + I2 + A1*A2eij + A1A2*e−ij

Next we can write down: A1A2 = (I1I2)½ and
eix = cos x + isin x and correspondingly e-ix = cos x - isin x, d.h. eij + e−ij = 2 cos j, because
eij + e−ij = cos j + isin j + cos (-j) + isin (-j) = 2 cos j

I = I1 + I2 + 2(I1I2)½. cos j