Accurately quantifying the superposition state of two different Laguerre–Gaussian modes with single intensity distribution measurement

We develop a method to quantify the superposition state of two different Laguerre–Gaussian modes. By analyzing the characteristics of the intensity distribution obtained in a single measurement, including the petal number, the position and value of the extremum intensity, one can quantify the angular momentum index, the radial node index and the superposition coefficient simultaneously. Experimentally, we measure a series of superposition states, whose angular momentum index ranges from −47 to 53, radial node index from 0 to 3 and superposition weight from 0.1 to 0.9. The average trace distance and the mean fidelity of these states are lower than 0.053 ± 0.001 and higher than 0.982 ± 0.002, respectively. Our method can further obtain the superposition coefficient compared with previous mode verification ones and can reduce the number of measurement settings compared with the traditional quantum state tomography, thus more applicable in practice.