- #1

PhysicsRock

- 117

- 18

- Homework Statement
- Calculate the diffraction pattern for a 2D Fraunhofer-diffraction, where monochromatic and coherent light of wave length ##\lambda## (wave number ##\vec{k}=2\pi / \lambda \hat{n}##) falls onto a wall with two slits, located in the regions where ##9d \leq |x| \leq 10d##. Both slits span infinitely in the ##y##-direction.

- Relevant Equations
- ##\displaystyle E(k_x,k_y) \propto \int_{-\infty}^\infty \int_{-\infty}^\infty a(x,y) e^{-i(k_x x + k_y y)} dx dy##

##I(k_x,k_y) \propto E^2(k_x,k_y)##

My issue here is the fact that the slits are supposed to infinite in the ##y##-direction. With what's given in the assignment, I'd define the apparatus function ##a(x,y)## as

$$

a(x,y) = \begin{cases} 1 & , \, ( 9d \leq |x| \leq 10d ) \wedge (y \in \mathbb{R}) \\ 0 & , \, \text{else} \end{cases}

$$

Plugging this into the Fourier transform and only considering the ##y##-part of it yields

$$

F(k_x,k_y) \propto \int_{-\infty}^\infty e^{-i k_y y} dy.

$$

One recognizes this as the integral representation of the delta-distribution, with a conventional factor of ##2\pi##. That would mean that

$$

F(k_x,k_y) \propto 2 \pi \delta(k_y).

$$

I'm unsure whether this is what is to be expected or not. The interpretation would be that there is a single sharp peak when ##k_y## is not 0, and if it is, the ##x##-part takes over and results in an oscillation, as I would expect.

$$

a(x,y) = \begin{cases} 1 & , \, ( 9d \leq |x| \leq 10d ) \wedge (y \in \mathbb{R}) \\ 0 & , \, \text{else} \end{cases}

$$

Plugging this into the Fourier transform and only considering the ##y##-part of it yields

$$

F(k_x,k_y) \propto \int_{-\infty}^\infty e^{-i k_y y} dy.

$$

One recognizes this as the integral representation of the delta-distribution, with a conventional factor of ##2\pi##. That would mean that

$$

F(k_x,k_y) \propto 2 \pi \delta(k_y).

$$

I'm unsure whether this is what is to be expected or not. The interpretation would be that there is a single sharp peak when ##k_y## is not 0, and if it is, the ##x##-part takes over and results in an oscillation, as I would expect.