Mathematical Optics

Exercise 5.23 Prove that a beam that satisfies Equation 5.97 is a spiral beam. Write the expression for this beam in far field.

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Since any field amplitude F(r) can be represented as a linear combination of orthonormal modes ${\mathcal{L}}_{m,n}(\mathbf{r})$, it can also be written as a sum of the spiral beams with the same velocity, for example, for v = 1, as

$F(\mathbf{r})=\sum _{m,n=0}^{\mathrm{\infty }}{f}_{m,n}{\mathcal{L}}_{m,n}(\mathbf{r})=\sum _{n=0}^{\mathrm{\infty }}{\mathrm{\Psi }}_n(\mathbf{r}|1),$

where Ψ(r|1) is given by Equation 5.97 and f_m,n = c_m,n.

The velocity of rotation is defined from the indices of any pair of modes ${\mathcal{L}}_{m_0,n_0}(\mathbf{r})$ and ${\mathcal{L}}_{m,n}(\mathbf{r})$ in the composition 5.94 as a ratio of the differences of the eigenvalues of these modes for symmetric FrFT and image rotator

$\upsilon =\frac{(m+n)-(m_0+n_0)}{(m-n)-(m_0-n_0)}=\frac{k}{l},$

FIGURE 5.17 Evolution of the beams ${\mathcal{L}}_{0,0}(\mathbf{r})+{\mathcal{L}}_{0,2}(\mathbf{r})$ (a) and ${\mathcal{L}}_{0,0}($