Question

3. Interference on a screen (40 pts) Consider two plane wave beams propagating in free space. They are identical except for the fact that they are misaligned with respect to each other by an angle 0, and are shined onto a screen. screen (a) If each beam has an amplitude Eo, write down an expression for the electric field of each plane wave in terms of Eo, the wavevector magnitude k, and the angular frequency w (using the coordinate system shown above). Use complex exponential notation. (15 pts) (b) Assuming the screen is at z=0, find the total intensity at the screen. (15 pts) (c) What is the average value of the intensity on the screen over all y? (5 pts) (d) What is the periodicity of the intensity on the screen in the y direction, in terms of 0 and ho? (5 pts)

Answer 1

3.) a.) consider the were rectors for two wares are ty of the - there Ri= k i t t = k[ loso 2 + sin 0 }] C (red) (blue) - The electric field for each plane here is Ē = Ēo enp [ilwt - R; ')} E Eo empfi (wt - F . %) ] Heu ñ= zêr yŷ Eoz Eo b) Total electric field at a=0 is Ecote En + E Es exp liwt) (exp(-i kz) + exp (ik26010 aikysin o)] - Eo exp (int) [ 1 + exp (-i ky shino) 230

: x دما Intensity : Eloto Eit tot - Eo?emp (int) exp(-owt) [1+ expl-i ky shrine) [1+ exp (ity shim)] 66[1 + 1 + eup (-i ky svo) + exp liby sis)] - to [a +260 (ky so)] [ •: exp(ix) +exfl-iso .2E [1+ loslky smo)} I : A to los? Ak y sino) – c.) we know that arecage value of Loso is zero. aresage value of los (ky sino) = 0 over all y so energe internity is 26 d.) from © we see that intensity is maximum at 0=0 with value 4E0². He will get some intensity then a ky sind a m there me Integu. considu at y, we have with maxima & aty (htigth marin se ky, sm O. MTť ky, tm @ = (m+1)T subtracting the two equalities ksino (4=Y) = 21 49-473 093 om - 2. 2 211 m 20 mo K Smo

So at this width on the screen we get the periodicity of the intensity.