Question

Suppose two plane waves of light with electric field amplitudes E1 and E2 arrive at a detector at time t having travelled different distances z1 and z2. Assume the field vectors have the same polarization direction P. The fields of the two plane waves at the detector individually are Epcos(ω-ka) and E2pcos(ot-kz2) . Here the wavenumber is k = 2πα and the angular frequency is ck 2nclv, where A is the wavelength, c is the speed, and is the frequency of the light. (a) What is are the time-averaged intensities 11 and I2 of the two plane waves? (b) If the two waves are coherent, we find the total field by summing the two fields. What is an expression for the time average of the intensity lotal of the resulting wave field? (c) Sketch a plot of the total intensity as a function of the difference in distances z2 zi- (d) In the plot, what are expressions for the maximum Itotal and minimum Itotal"" values and min what is the visibility Vis El total max l total mln )/(1 total max + 1 total mun ) ?

Answer 1

Given information E_1 = E_1 cos(w_t - K z_1) beta E_2 = E_2 cos(w t - k z_1) p E_2 = e_2 cos(w t - k z_2) p a) what is the time - averaged intensities of l_1 and l_2 of two plane wave? time average of any function f(t) an be given by f(t) > T = 1/T integral_0^T f(t') dt' now let us first see the expression for the intensity 1 in general from the electric field E which is given by I = E. E = E^2 Then the time averaged intensity is given by LI > T = 1/T integral_0^T E^2(t') dt' Now the time - averaged intensity is given by < I_1 > T = 1/T integral_0^T E_1^2 cos^2(wt^1 - k z_1)dt' = 1/T e_1^2 integral_0^T cos^2(wt' - k z_1) dt' \r\n 1/T E_1^2 integral_0^T[1 + cos^2 2(wt - k z_1)]/2 dt' E_1^2/2 T integral_0^T dt' + integral cos^2(wt') - k z_1)] dt' = E_1^2/2 T{T + 0} < I_1 > T = E_1^2 ---- Similarly the time - averaged intensity