Diffraction at apertures with arbitrary angle to wavefront Every physics book I know explains diffraction (e.g. at slits, gratings and circular apertures) in a way which assumes that the light wave impinges perpendicularly on the aperture. Would there be any difference if that was not the case (i.e. if I have a slit and adjust a laser such that it impinges with a certain angle instead of pointing directly towards the slit)? I know that diffraction can usually be explained using Huygens' principle but I am not sure what to get from that in this specific case. Especially, I am wondering if the time difference between the wave's arrival at the individual ends of the aperture would have any effect (I guess so but I could not tell how that would look like).  
 A: In Fraunhofer diffraction, the farfield pattern is proportional to the spatial Fourier transform of the input field, so the tilt on the input field simply translates the diffraction pattern transversely.
A tilt corresponds to multiplying the input field by $\exp(i\,\vec{k}_0\cdot\vec{r})$ where $\vec{k}_0$ is the wavevector showing the nominal propagation direction of the whole beam (otherwise put: it would have the same direction as the chief ray in a ray analysis of the beam)and $\vec{r}$ is the position in the input aperture. If the beam is untilted, $\vec{k}_0$ is normal to the aperture, and therefore $\exp(i\,\vec{k}_0\cdot\vec{r}) = 1$ everywhere in the aperture.
Recall the Fourier shift lemmas that if $\mathfrak{F}_{\vec{k}} f(\vec{r}) = F(\vec{k})$ then $\mathfrak{F}_{\vec{k}} f(\vec{r}+\vec{r}_0) = \exp(i(\vec{k}\cdot\vec{r}_0))F(\vec{k})$ and that $\mathfrak{F}_{\vec{k}} [\exp(i(\vec{k}_0\cdot\vec{r}))f(\vec{r})] = F(\vec{k}-\vec{k}_0)$. Hence you can see that the tilt is simply shifting the diffraction pattern sideways, as claimed. 
More simple mindedly: if you trace the tilted chief ray to the plane where the diffraction pattern forms, its intersecton with this plane is where the centre of an untilted field's diffraction pattern is shifted to.
A second order effect is the nonuniform vignetting arising from the tilt. In Fraunhofer diffraction we can model this effect by saying that the aperture changes effective shape to the projection of the aperture onto the beam's transverse plane, i.e. onto a plane at right angles to the chief ray. Therefore a circular aperture of radius $r$ becomes an effectively elliptical aperture of major and minor semiaxis lengths $r$ and $r\cos\theta$, where $\theta$ is the tilt angle.

Further Question

Am I right assuming that this effect could then be transferred to a slightly focussed (perpendicular) beam using superposition? So let's say I have a laser beam which is not well collimated but rather slightly converging and I adjust it such that it points onto a slit with the beam axis being perpendicular to the plane of the slit. Would the diffraction pattern then be kind of "smeared" as one can regard the beam as a superposition of differently tilted individual rays which each cause a slightly different pattern? 

In principle you could model a focussed beam like this: it is a good conceptual way to think about the "smearing" speak of: you would split the focusing field up into a plane wave superposition. Suppose it is represented by a ray bundle converging on the aperture; each ray $j$ in the focusing bundle then represents a plane wave with wavevector $\vec{k}_j$ (here $|\vec{k}_j| = k = 2\pi/\lambda$ in the direction of the ray. You then get a tilt phase  $\exp(i\,\vec{k}_j\cdot\vec{r})$ across the aperture. You would then take the Fourier transform of the aperture function to find its projection onto the screen, and sum up all the effects for all the rays. Naturally,  you would add these coherently; you add the Fourier transforms then take their square magnitude to find the fringe pattern. In practice there is an easier way to do this in one hit: add up all the  $\exp(i\,\vec{k}_j\cdot\vec{r})$ tilts at the aperture then take one Fourier transform of the sum. Fourier transforms, and general diffractions are linear. A focusing beam to first order it can be modeled by $\exp\left(i k \,\kappa\frac{|\vec{r} - \vec{r}_0|^2}{2}\right)$ where $\vec{r}$ is the transverse position in the aperture, $\vec{r}_0$ the position in the aperture of the beam centre and $\kappa$ the wavefront curvature, so this is the function you would multiply the otherwise uniform aperture by.
