Finding Fourier transform of a periodic lattice

Question

I have a 1D periodic lattice with 'p' components (unit cells). I want to take a discrete Fourier transform in space to get a reciprocal space, where I will have $k$ (reciprocal vector) dependence.

I generate the 1D periodic lattice with "p" components by this MWE:

p = 10;

a = Table[0, {i, 1, p}, {j, 1, p}];

For[i = 1, i <= p, i++,
  

  For[j = 1, j <= p, j++,
   a[[i, j]] = 
     If[(Abs[i - j] == p - 1 \[Or] 
         Abs[i - j] == 1) \[And] (j <= p \[And] i <= p), 1, 0];
   ];
  ];

AdjacencyGraph[a]

Now, I would like to generate the reciprocal lattice from the above, where each point $a[i] = \sum_{k} e^{ik}\, a[k] $.

reciplatA = Fourier[a, FourierParameters -> {0, 2 \[Pi]/p}]; 

But I don't think I want this form. It is supposed to be a diagonal matrix, which isn't the case. Thus, how can I have some Fourier variable conjugate to the space variables, i.e., $k$ appearing after the transform.
How can we obtain such a form?
Can we extend this to 2D, where we will have two Fourier variables $\{k_1, k_2\}$?

Answer 1

Here are two most simple examples I can think of. I choose a potential that is periodic and symmetric around the first point in order to get a real FFT.

Note that the output from FFT is a bit confusing. It is a list with coefficients of:{ DC component, increasing positive frequencies, decreasing negative frequencies}. Look it up in the manual or a book.

Here is a 1 dim example with period 10:

d = ConstantArray[0, 100];
Do[d[[i]] = 1, {i, 1, 100, 10}]
fft1 = Fourier[d] // Chop

And here a 2 dim example:

d = ConstantArray[0, {100, 100}];
Do[d[[i, j]] = 1, {i, 1, 100, 10}, {j, 1, 100, 10}];
Fourier[d] // Chop