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We have an experiment, where we have to calculate various parameters of helix using its diffraction pattern.

The experiment goes in three steps :

  • First we observe the diffraction pattern due to a single thread and measure the wavelength of the light used. enter image description here

  • The diffraction pattern of helix can be thought of as two sets of parallel wires kept at an angle to each other. This gives us diffraction minima enveloped over minima due to interference due to these wires. enter image description here

  • I'm stuck at the third step. There we have a double helix, but we can't figure out how to parameters of both the helices using the same analogy as above. enter image description here Can anyone suggest something please?

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When you have a function that is duplicated and shifted relative to each other, it is like you have convolved the function with a pair of Dirac deltas. $$ f({\bf x})\star [\delta(z-z_0)+\delta(z+z_0)] $$ In the Fourier domain (far-field) this implies that you multiply the spectrum of that function with a cosine function, $$ 2 F({\bf k}) \cos(k_z z_0) $$ which is the Fourier transform of the pair of Dirac deltas. The period of the cosine function is inversely proportional to the separation between the two Dirac deltas. Perhaps this can help you to interpret what you would see with the double helix.

For a bit more detail of Fourier transforms and Dirac delta functions, the Fourier transform of a shifted Dirac delta is an exponential function $$ {\cal F}\{\delta(x-x_0)\} = \int_{-\infty}^{\infty} \delta(x-x_0) e^{i kx} dx = e^{i kx_0} . $$ Hence, $$ {\cal F}\{\delta(x-x_0)+\delta(x+x_0)\} = e^{i kx_0}+e^{-i kx_0} = 2\cos(k x_0) . $$

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  • $\begingroup$ Can you please elaborate on that delta function part. I'm not well versed with them. $\endgroup$ Commented Jan 30, 2018 at 13:23
  • $\begingroup$ Hope this helps. $\endgroup$ Commented Feb 1, 2018 at 12:50

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