consider 2 monochromatic waves with wavelength $\lambda$ that travel different paths with path difference $s$ that meet.

• If $s=n\lambda \quad n=\Z$ then the waves are in phase and we get constructive interference ( phase difference $\phi = \frac{2 \pi s}{\lambda}= 2 n \pi$ )
• if $s=(n+\frac{1}{2})\lambda \quad n=\Z$ then the waves are out of phase and we get destructive interference ( phase difference $\phi=(2n+1)\pi$ )

Huygens’ principle

Every point on a primary wavefront acts as a source of secondary wavelets such that the wavefront at some later time is the envelope of these wavelets.

• Huygens’ principle is empirical thus it is approximative and provides only a qualitative description

Young’s double-slit experiment

Consider a monochromatic plane wave incident upon an opaque barrier containing two very narrow slits $S_1$ and $S_2$ separated by a distance $a$ we can derive the following properties:

• We get bright peaks when:

$$ a \sin\theta = n \lambda \quad n=\Z $$

• We get dark fringes when:

$$ a \sin\theta = (n+\frac{1}{2}) \lambda $$

• The amplitude of the wave at a point P on the screen is

$$ \begin{aligned} R&= A \left[ \cos(\omega t - k l_1) + \cos(\omega t - k l_2) \right]\\ &= 2A \cos\left( \omega t -k \frac{l_1+l_2}{2}\right) \cos\left( k \frac{l_1-l_2}{2}\right) \end{aligned} $$

• Which using the approximations $l_1 \sim l_2 \sim L$ and $(l_1+l_2)/2 \simeq L$ we can write the intensity:

$$ I=R^2 = 4A^2 \cos^2\left( \omega t -k L\right) \cos^2\left( k \Delta l/2\right) $$

• We can get the path difference

$$ I=I_0 \cos^2 \frac{k a\sin\theta}{2} \quad \mathrm{where} \quad I_0=2A^2 $$