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Diffraction experiment with waves |
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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/ |
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
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