As well known, we can prove Poincare duality in terms of morse theory. (By comparing two chain complexes obtained from two morse functions, $f\colon M\to \mathbb{R}$,$-f\colon M\to \mathbb{R}$ for smooth closed manifold $M$)

Of course, I can prove Alexander duality theorem by using usual Lefschetz-Poincare duality theorem and some excision arguments.

But, I want to prove Alexander duality theorem by using a morse function defined on $S^{n}-X$, for a submanifold $X$ of $S^n$.

Are there any references or proofs about this?