## Abstract

We demonstrate a simple approach, using digital holograms, to perform a complete azimuthal decomposition of an optical field. Importantly, we use a set of basis functions that are not scale dependent so that unlike other methods, no knowledge of the initial field is required for the decomposition. We illustrate the power of the method by decomposing two examples: superpositions of Bessel beams and Hermite-Gaussian beams (off-axis vortex). From the measured decomposition we show reconstruction of the amplitude, phase and orbital angular momentum density of the field with a high degree of accuracy.

© 2012 OSA

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### Equations (13)

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(1)
$$u(r,\varphi )={\displaystyle \sum _{p,l}{{c}^{\prime}}_{p,l}{R}_{p,l}(r)}\mathrm{exp}(il\varphi ),$$
(2)
$$u(r,\varphi )={\displaystyle \sum _{l}\left[{\displaystyle \sum _{p}{{c}^{\prime}}_{p,l}}{R}_{p,l}(r)\right]}\mathrm{exp}(il\varphi )={\displaystyle \sum _{l}{c}_{l}(r)}\mathrm{exp}(il\varphi ),$$
(3)
$${c}_{l}(r)={\displaystyle \sum _{p}{c}_{p,l}{R}_{p,l}(r)}={a}_{l}(r)\mathrm{exp}\left[i\Delta {\theta}_{l}(r)\right].$$
(4)
$${c}_{l}(r)=\frac{1}{2\pi}{\displaystyle \underset{0}{\overset{2\pi}{\int}}u(r,\varphi )\mathrm{exp}(-il\varphi )d\varphi}.$$
(5)
$${U}_{l}(0)=\frac{\mathrm{exp}(i2kf)}{i\lambda f}{\displaystyle \underset{0}{\overset{\infty}{\int}}{\displaystyle \underset{0}{\overset{2\pi}{\int}}{t}_{l}(r,\varphi )u(r,\varphi )rdrd\varphi}}.$$
(6)
$${t}_{l}(r,\varphi )=\{\begin{array}{cc}\mathrm{exp}(-il\varphi )& R-\Delta R/2<r<R+\Delta R/2\\ 0& otherwise\end{array}.$$
(7)
$$\begin{array}{c}{\tilde{U}}_{l}(R,0)=\frac{\mathrm{exp}(i2kf)}{i\lambda f}{\displaystyle \underset{0}{\overset{\infty}{\int}}{\displaystyle \underset{0}{\overset{2\pi}{\int}}{t}_{l}(r,\varphi )u(r,\varphi )rdrd\varphi}}\\ \approx \frac{k\mathrm{exp}(i2kf)}{if}{c}_{l}(R)R\Delta R.\end{array}$$
(8)
$${c}_{l}(R)=\frac{if}{R\Delta Rk\mathrm{exp}(i2kf)}{\tilde{U}}_{l}(R,0).$$
(9)
$${a}_{l}(R)=\left|{c}_{l}(R)\right|=\frac{f}{R\Delta Rk}\sqrt{{I}_{l}(R,0)}.$$
(10)
$$\begin{array}{c}{\tilde{I}}_{l}(\Delta {\theta}_{l})={\left|{c}_{l}(R)+g\right|}^{2}\\ ={a}_{l}^{2}(R)+{\left|g\right|}^{2}+2{a}_{l}(R)\left|g\right|\mathrm{cos}[\Delta {\theta}_{l}(R)-\alpha ].\end{array}$$
(11)
$$S=\frac{{\epsilon}_{0}\omega {c}^{2}}{4}\left[i\left(u\nabla {u}^{*}-{u}^{*}\nabla u\right)+2k{\left|u\right|}^{2}\stackrel{\u2322}{z}\right],$$
(12)
$${L}_{z}=\frac{1}{{c}^{2}}{\left[r\times S\right]}_{z}.$$
(13)
$$u(r,\varphi )={A}_{m}{J}_{m}(kr)\mathrm{exp}(im\varphi )+{A}_{n}{J}_{n}(kr)\mathrm{exp}(in\varphi )\mathrm{exp}(i\Delta \theta ).$$