# Are there any more physically representative analytical expression for a slit or edge than a step-function? What about ${\rm erf}(x)$ for example?

Historically slits have been invaluable in teaching, research, and theory validation in both electromagnetic and quantum mechanics but conceptually they differ from what we can actually build based on the physical properties of matter because the canonical slit perfectly absorbs an incident wave in an infinitely thin layer without reflection or induced phase shifts.

This simple implementation also results in discontinuities in the amplitude at the slit edge. We know this is wrong but we still get pretty good results when matching up the resulting calculated pattern produced from a hard-edged infinitely thin theoretical binary slit that just multiplies the incoming wave by either unity inside the slit opening or zero outside.

Question: Have more sophisticated analytical models of slits been suggested that will function similarly but take steps to be more physically realistic in terms of finite thickness and reduced discontinuity?

Just as an illustrative example $$\frac{1}{2} \text{erf}\left(\frac{x+1}{\sigma}\right) - \frac{1}{2}\text{erf}\left(\frac{x-1}{\sigma}\right)$$ looks a little "softer" than a pair of step functions, but I don't know if it is any better or worse in terms of the wave mechanics.

Notes:

1. As pointed out in comments I'm really asking about modeling single edges; this could apply to the edges of rectangular or circular apertures, or even diffraction from a single straight edge.
2. Answers that addresses either an electromagnetic wave or a matter wave (e.g. atoms) are welcome.
3. I've asked about analytical models for slits rather than constructs used in finite element analysis, but there may be something to be learned from those impedance-matched constructs.
4. Answers to In the double slit experiment what, exactly, is a slit? don't go far enough to answer this question.
• You get interference patterns from single edges. Photons directed toward a single edge will create an interference pattern unlike a slit pattern. The spacing is not equal but diminishes getting smaller and smaller farther away from the shadows edge. A slit is just two single edges facing each other and the two single edge patterns projected on the screen overlap each other and convolute to create an equally spaced slit interference pattern. Jun 7, 2020 at 2:03
• @BillAlsept you are of course right, I'm really asking about the modeling of edges and what I'm looking for could apply to rectangular or circular apertures as well. I've updated the question to make that clearer, thanks!
– uhoh
Jun 7, 2020 at 2:06
• Well, carefully measure a diffraction pattern and take the Fourier transform of it. Jun 7, 2020 at 2:59
• @JonCuster that's a great idea! You can do that in the far field but it would not then apply to a near-field experiment (e.g. microwaves on slotted sheet metal) unless you measure phase as well and do a complex transform. This sounds fun! Now all I need is a microwave oven and an oscilloscope...
– uhoh
Jun 7, 2020 at 3:01
• @uhoh For the purposes of modeling the actual shape of the edge, you can measure the diffraction pattern in the far field (possibly in a separate apparatus), and then use the data you gathered about the shape of the edge to make predictions about what would happen in the near field, right? Sep 5, 2020 at 3:32

A reasonable mathematical model for a gradual transition from one state to another is the $$\tanh$$ function.
Thus $$T(x) =\frac{1}{2}\left( 1 + \tanh(x/t)\right)$$ smoothly transitions from $$T=0$$ when $$x<0$$ to $$T=1$$ for $$x>0$$, with a characteristic "width" for the transition of $$\pm t$$ about $$T=0.5$$ at $$x=0$$. A sharp edge is recovered by allowing $$t \rightarrow 0$$. The plot below shows $$T(x)$$ for $$t=1$$. An alternative is the logistic sigmoid. $$T(x) = \frac{1}{1 + \exp(-2x/t)}$$ where $$t$$ has a similar definition. The plot below shows this function for $$t=1$$. 