Fig. 1. Schematic diagram of the measurement of the coherence phase spectrum for a partially coherent fractional vortex beam. (a) Source plane with unknown topological charge. (b) Focal field. (c) Full source coherence information is restored. (d) Coherence phase spectrum by coherence phase is decomposed.

Fig. 2. Experimental setup. (a) Generation of the partially coherent fractional vortex beams with different amplitudes and coherence lengths. BE, beam expander; L1, L2, L3, L4, thin lenses, with focal lengths of 100, 150, 300, and 300 mm, respectively; RGGD, rotating ground glass disk; BS, beam splitter; SLM, spatial light modulator; USAF (1951USAF resolution test chart) acts as an object; CCD1, charge coupled detector (ECO445); CCD2, electron-multiplying CCD. The first line of (b) and (c) is the hologram patterns loaded to the SLM for generating a fractional vortex beam with l=2.5. The second row of (d) and (e) is the experimental results of the focused fractional vortex beams (source intensities are inserted). PV, plane wave vortex beam; EGV, elliptic Gaussian vortex beam. LG, Laguerre–Gaussian vortex beam. High C, high coherence; δ0=1.5  mm. MC, medium coherence; δ0=0.4  mm. LC, low coherence; δ0=0.25  mm.

Fig. 3. Reconstruction of the source coherence phase. (a1) Focal average intensity, (a2) cross-spectral density amplitude, and (a3) cross-spectral density phase recorded in experiment. (a4) Source average intensity, (a5) cross-spectral density amplitude, and (a6) cross-spectral density phase reconstructed by inverse propagation to the source plane. (b1)–(b6) Cross-spectral density amplitude and cross-spectral density phase in the source plane for different amplitudes. (c1)–(c6) Cross-spectral density amplitude and phase in the source plane for different coherence lengths.

Fig. 4. Coherence phase spectrum. Theoretical and experimental coherence phase spectra with (a)–(c) different amplitude envelopes and (d)–(f) different degrees of coherence.

Fig. 5. Accuracy and applicability. (a) Accuracy verification; (b), (c) integer and larger topological charge measurements.

Fig. 6. Application in optical encryption. Conceptual illustration of the optical information encoding scheme using fractional vortex beams.