For the purposes of this problem, we assumethat two long parallel slits extending along thezaxis (out of the plane shown in the figure) are separted by a distanced. they are illuminated coherently, that is, in phase, by light with a wavelengthlambda, for example by a laser beam polarized in thezdirection. (lacking a laser, young used an intense source diffracted bya slit to produce coherent illumination of his double slits.)

the key point is that the electric field far downstream from the slit (e. g., at a large positivexvalue) is the sum of the electric fields emanating from each of the twoslits. hence, the relative phase of these electric fields at someobservation pointodetermines whether they add in phase (constructively) or out or phase (destructively).view figure

to refresh your memory about traveling waves, the electric fielde(x, t)that is incident on the double slits from the left is a function ofxandt. let us assume that it has amplitudee_left. we will also assume a cosine trigonometric function with the arbitrary phase set equal to zero (i. e., at the pointx=0,\; t=you find thate=e_{\rm left}). then

e(x, t) = e_{\rm left}\cos{ \left[ \frac{2 \pi} {\lambda} \left(x - ct \right) \right]}.

the argument of the cosine function is the phasephi(x, t). it can be written as\phi(x, t)=kx-\omega t, whereomegais the angular frequency (\omega = 2\pi f) andkis thewave numberdefined byk=2\pi/\lambda. the phase increases by2\pieach time the distance increases bylambda. moreover, the phase is constant for an observer moving in the positivexdirection