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With Fourier-Series Expansion, we can write a function as sum of many non-repating different frequncied different amplituded sine and cosine functions.

Lets assume we know electric-field and magnetic-field representation function of Hydrogen atoms or Iron atoms as a periodic-lattice (If we cannot know, lets take periodic-Dirac-Delta).

Question: how many sine+cosine wave generators do we need to produce such imitation for at least xy plane with minimal erroer(lets say %0.1 about electron-orbit's uncertainity) ? (ofcourse electromagnetic waves are transverse sinusoidal) Very narrow angled generators targeted at same points so they would make a spot of super-positioned waves acting like Iron lattice or whatever em-field needed maybe (even a shield against big objects, with enough power source and strong generators, like in the Star-Trek realm?).

enter image description here

like ion cores in the lattice as a barrier for nano-particles.

Thanks.

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    $\begingroup$ Any non-repeating function Fourier transforms to a continuous function not a discrete sum, so I don't think your question has any useful answer. $\endgroup$ Commented Jun 27, 2013 at 15:16
  • $\begingroup$ @John Rennie: Yes non-repeating, yes continuos(periodic Dirac-Delta) $\endgroup$ Commented Jun 27, 2013 at 15:41
  • $\begingroup$ Have a look at falstad.com/fourier. If you select triangle and rectify you get something like the example you give. $\endgroup$ Commented Jun 27, 2013 at 17:06
  • $\begingroup$ Do you mean you want to use something like many-many-many lasers to generate your sines? If yes, it'd be quite difficult because you have to make sure your EM waves overlap with very small angles between them. Also, you'd have to make them purely standing waves, otherwise you'll not have stationary function. $\endgroup$
    – Ruslan
    Commented Aug 4, 2013 at 19:28
  • $\begingroup$ Maybe focusing through a real thin lattice possible? $\endgroup$ Commented Aug 4, 2013 at 21:03

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Ok, let's take a gaussian profile, and we'll repeat it every $2\pi$ to turn it into a function for which we can calculate the Fourier coefficients. The resulting function looks like this:

Function

I've written a spreadsheet to calculate the Fourier components for this function and use the components to recalculate the function (I can put the spreadsheet somewhere downloadable if you want).

If I take the first 10 components the fit is close to perfect (with more than 10 components you can't see the difference between the fit and the original function):

10 components

With the first 6 components you can clearly see the difference:

6 components

and with only 3 components the approximation is fairly poor, as you'd expect, though you can already see the central peak developing:

3 components

I'm still not entirely sure what you're asking, but this should give you an idea of how many terms you need to get a reasonable approximation to your target function.

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  • $\begingroup$ So ten different wave generator arrrays(or matrices) are enough for less than %1 error? $\endgroup$ Commented Aug 5, 2013 at 11:33
  • $\begingroup$ Then just 20-30 generators would be better than %0.1 error. $\endgroup$ Commented Aug 5, 2013 at 11:40
  • $\begingroup$ With 10 terms the maximum is 1.8% (at the peak). To get less than 0.1% difference requires only 15 terms. $\endgroup$ Commented Aug 5, 2013 at 11:42
  • $\begingroup$ Note that a gaussian is a fairly easy function to fit. If you chose a different function this would affect the number of terms required. If you choose a function with discontinuities, or discontinuities in the gradient, it will take a large number of terms to get the error small at the discontinuity. $\endgroup$ Commented Aug 5, 2013 at 12:03
  • $\begingroup$ Can you do same for a dirac-delta ? $\endgroup$ Commented Aug 5, 2013 at 12:06

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