(Solved): Problem 1. Fraunhofer diffraction of a double slit. You learned Young's double slit experiment in ba ...
Problem 1. Fraunhofer diffraction of a double slit. You learned Young's double slit experiment in basic physics course. Now you have the theoretical tool to calculate the diffraction pattern. Let's do it step by step. (a) (1 point) Consider a pair of silts locating at a plane of
z=0. The distance between them is
d. The width of each slit is infinitely small. In this case, the scalar field at the slit is given by
u_(0)(x_(0),y_(0))=(1)/(2)\delta (x-(d)/(2))+(1)/(2)\delta (x+(d)/(2)).Please calculate the far-field diffraction pattern. *Hint: Follow the same procedure we did on page 6-16. Use Shift theorem. (b) (1 point) Please show that the constructive interference occurs at
dsin\theta =m\lambda where
mis an integer. (c) (1 point) If you introduce a
\pi phase shift to one of the slits, how does the far-field diffraction pattern become? That is, please calculate the far-field diffraction pattern for
u_(0)(x_(0),y_(0))=(1)/(2)\delta (x-(d)/(2))+(1)/(2)e^(i\pi )\delta (x+(d)/(2)).(d) (1 point) Now consider the slits have a finite width of
b.
u_(0)(x_(0),y_(0))=(1)/(2)\Pi ((x-(d)/(2))/(b))+(1)/(2)\Pi ((x+(d)/(2))/(b)).please calculate the far-field diffraction pattern. *Hint: Notice that
\Pi ((x-(d)/(2))/(b))=\Pi ((x)/(b))ox\delta (x-(d)/(2)). By using the convolution theorem, you can use the result of the single silt diffraction to calculate the double slit diffraction. (e) (1 point) Please plot the intensity distribution of the result of (d) at a
z=1mas a function of x. You can use Matlab or any software. Assume
d=10\mu m,\lambda =0.5\mu m, and
b=50\mu m. Normalize the peak intensity to be 1 .
