Single slit diffraction / wave propagation

[gcad]

I have a couple of questions about single slit diffraction. It's not a homework exercise - just a conceptual problem. Here it is (with some introductory comments)...

I understand that the diffraction pattern from a single slit is the modulus squared of the Fourier transform of the aperture. We have all seen the picture of the slit and the resulting |sinc|^2 profile. My question is in regards to what the position/location of these fringes mean in terms of spatial frequency. I'm wondering exactly what these spatial frequencies correspond to.

I'll have a go - Is it that the sum of plane waves having different spatial frequencies (defined by the diffraction pattern) will, when summed up, equal the object (aperture)?

If so, what spatial frequency does the first bright spot away from the central spot represent? Is the wave vector the addition of the original (defined by the incident propagation direction) and the displacement vector (defined by the transverse position along the diffraction pattern)? OR does the first bright spot simply represent the inclusion of a wave with spatial frequency defined by the displacement vector, i.e. a wave traveling perpendicular to the incident wave?

comments please!
apologies for the long-windedness! :-)

 

[Andy Resnick]

There's a few slight misconceptions that need to be addressed first:

1) the far-field diffraction *amplitude* pattern is the Fourier transform of the field amplitude at an aperture.

2) spatial frequency is the Fourier Transform pair (dual?) to position. A single spatial frequency can be interpreted as a plane wave propagating in a particular direction. In particular, f = x/lz, where f is the spatial frequency, x the coordinate measured at the aperture, l the wavelength of light, and z the propogation distance. That expression can be turned around to measure spatial frequency in terms of angle from the aperture, but I can't seem to find it right now.

So, the central spot (on axis) corresponds to a spatial frequency of zero, and as you move away from the optical axis, the spatial frequency increases.

And, recall the above definition is for field *amplitudes*, not intensities. There's a difference when taking the coherence of the light into account- the OTF for coherent light is the pupil function, but the OTF for incoherent light is the autocorrelation of the pupil function. Since most experiments are performed using highly coherent light, there's usually no confusion.

 

[astrokreng]

I am happy to address your questions about single slit diffraction and wave propagation. First, I would like to clarify that the diffraction pattern from a single slit is not the modulus squared of the Fourier transform of the aperture. It is actually the Fourier transform of the aperture, which is then squared to give the intensity pattern. This is a common misconception, but it is important to understand the difference between the two.

Now, to answer your question about the spatial frequencies in the diffraction pattern, they correspond to the spatial frequencies of the plane waves that make up the diffracted wave. In other words, the diffraction pattern is a representation of the spatial frequencies present in the diffracted wave. The first bright spot away from the central spot represents the inclusion of a plane wave with a spatial frequency that is defined by the displacement vector, as you mentioned. This means that the wave is traveling perpendicular to the incident wave, as you correctly stated.

In terms of the wave vector, it is indeed the addition of the original wave vector (defined by the incident propagation direction) and the displacement vector (defined by the transverse position along the diffraction pattern). This is because the diffracted wave is a combination of the incident wave and the waves that are diffracted at different angles due to the slit. The different spatial frequencies in the diffraction pattern correspond to these different angles of diffraction.

I hope this helps to clarify your understanding of single slit diffraction and the spatial frequencies involved. If you have any further questions, please don't hesitate to ask. As scientists, it is important to always question and seek a deeper understanding of the concepts we are studying. Keep up the curiosity!