Fig. 1. Conceptual setup and the requirement for the KK retrieval. (a) The signal field with a complex OAM spectrum is co-axially combined with a reference Gaussian beam. The intensity of their interferogram is recorded with a camera and is used for the spectrum retrieval. (b) The azimuthal trajectories of the signal (left) and the interferogram (right) in the complex plane. In order to meet the minimum phase condition, the trajectory must not encircle the origin. With the addition of a sufficiently large reference field (denoted by the dashed arrow), the interferogram satisfies the requirement, and thus the KK method can rigorously reconstruct the complex signal OAM spectrum.

Fig. 2. Experimental single-shot KK retrieval. (a)–(c) Retrieval process of an arbitrary OAM spectrum. (a) The intensity images of the reference (top left) and the interferogram (top right) are captured by a camera. By sampling along the dashed circles in the images, their azimuthal distributions are extracted (bottom). (b) The interferogram is then normalized and digitally upsampled, whose phase is extracted from the Hilbert transform of the logarithm of its amplitude (top). From the full field of the interferogram, the amplitude and phase of the signal are obtained and downsampled to the original sampling points (bottom). (c) Taking the Fourier transform of the signal field derives the normalized amplitude (top) and relative phase relation (bottom) of the OAM spectrum. An accuracy of 97.6% is achieved by comparing the retrieved and the target OAM spectra. (d), (e) Measured 2D bar charts of the OAM states for (d) the superpositions of mode indices from 1 to n (n=1,2,…,20) with equal amplitudes and in-phase relations; (e) the superpositions of mode indices n and 21−n (n=1,2,…,20) with equal amplitudes and a π/2 phase shift. The average retrieval accuracies in (d) and (e) are 98.9% and 96.0%, respectively.

Fig. 3. Measurements of complex OAM spectra. Left, the measured intensity images of the signal and the interferogram; right, the normalized amplitude and relative phase of the target and experimentally retrieved OAM spectrum. (a)–(d) Gaussian-shaped OAM spectra centered at 10th order with versatile OAM mode spacings and relative phase relations. (a) In-phase OAM spectrum with a mode spacing of 1. (b) Linear-phase OAM spectrum with a phase slope of 2π/3 and a mode spacing of 1. (c) In-phase OAM spectrum with a mode spacing of 3. (d) OAM spectrum with periodic Talbot phase [0,2π/3,2π/3] and a mode spacing of 1. (e) Fractional OAM mode with a topological charge of 9.5. The retrieval accuracy is also indicated for each case.

Fig. 4. KK retrieval at different CSPR levels. (a) The complex amplitude field of the signal used for the study [the same as the signal field in Fig. 3(a)]. (b) The azimuthal trajectories of the signal field and interferograms with in-phase and out-of-phase addition of the reference field. The trajectories of interferograms are exactly at the limit of the minimum phase condition, which correspond to the minimum required CSPRs of 5.1 dB (in-phase) and 11.3 dB (out-of-phase), respectively. (c) The accuracy of the KK retrieval at different CSPR levels and varying phases between the signal and reference fields. Below the CSPR threshold for arbitrary phases (11.3 dB, indicated by the dashed line), the retrieval performance varies with the phase, while above, the retrieval accuracy is approximately close to the unity for all the phases.

Fig. 5. Performance evaluation of the KK retrieval on random OAM spectra. (a), (b) Histograms of the retrieval accuracy of the KK method and the conventional Fourier method, measured on 1000 OAM spectra with random complex mode coefficients. (a) For an OAM measurement range from 1 to 20, the average and standard deviation of the KK retrieval accuracy are 95.6% and 1.2%, respectively. (b) For an OAM measurement range from 1 to 30, the average and standard deviation of the KK retrieval accuracy are 91.1% and 1.4%, respectively. The KK method shows superiority over the Fourier method in both cases.