Fig. 1. Schematic diagram of incoherent modal decomposition. Part-1: generation of a partially coherent beam with adjustable coherence width. The coherence width can be modified by shifting the spherical lens before the RGGD back and forth. Part-2: intensity monitor. The GSM beam propagates through a spherical lens system (f=300mm), and CCD1 is used to monitor the intensity distribution. Part-3: modal decomposition. A high-scattering object (USAF) is placed on the measurement plane, and CCD2 is used to obtain the diffraction pattern. The USAF is fixed on a translation stage, and it moves on the transverse plane to take diffraction patterns. The spherical lens system is replaced with (a) a cylindrical lens system and (b) a turbulent atmosphere system for the study of coherence entropy in complex media. The focal length of cylindrical lens is 150 mm. BE, beam expander; GAF, Gaussian amplitude filter; USAF, 1951 USAF resolution test chart; CCD, charge-coupled device; BS, beam splitter; and CL, cylindrical lens.

Fig. 2. Basis adjustment for modal decomposition. (a) A GSM beam in the source plane and (b) the elongated beam spot after going through a cylindrical lens system. Using the symmetric HG basis, mode-weight results in panels (d) and (e) are calculated for (a) and (b), respectively. (f) Basis after applying the transmission matrix on panel (c), which helps recover the mode-weights. (g)–(j) Mode-weight distributions of a GSM beam propagating through a cylindrical lens for various coherent widths. The orange bars correspond to the mode-weight distribution before adjusting the orthogonal basis, and the green bars show the mode-weights after adjusting the orthogonal basis. Here, only the main part (5×5) of the mode-weight distribution (8×8 in total) is displayed.

Fig. 3. Coherence entropy of a GSM beam passing through the turbulent atmosphere. The mode-weights for a GSM beam propagating through (a)–(c) low turbulence (MW1) and (d)–(f) high turbulence (MW2) atmosphere. (g)–(i) The differences in mode-weights (|MW1−MW2|) for two turbulence cases. A low-turbulence atmosphere corresponds to room temperature; a high-turbulence atmosphere is realized by a hot plate at 200°C. (j)–(l) Coherence entropy calculated from mode-weights. (k) and (l) Zoomed-in field of view of the medium-coherence and low-coherence cases. Illustrations in panel (j) show the two intensity deformations caused by a turbulent atmosphere. The deformation is less apparent for lower-coherence case. The blue, yellow, and red bars correspond to the mean value of coherence entropy, under low-, medium-, and high-turbulence atmospheres, respectively. The black error bar shows the standard deviation of 10 data sets.

Fig. 4. Performance improvement in OAM-based optical encryption and decryption through turbulent media using basis adjustment. OAM demultiplexing in free-space using (a) incoherent decomposition and (b) traditional multifocal array consisting of different integer vortices. (c) Wavefront distortion induced by the turbulent media. Chaotic OAM demultiplexing through turbulent media using (d) orthogonal modal decomposition and (e) traditional multifocal array. (f) Corrected OAM demultiplexing through turbulent media using mode adjustment for incoherent decomposition. (g) Encryption of a clock-style number string “215006.” (h) Experimental demonstration showing that the correct decryption may be achieved after applying mode adjustment in incoherent decomposition, while the traditional method yields highly inaccurate results.

Fig. 5. Channel robustness evaluation using the coherence entropy. Mode-weight distribution (a) in free space and (b) through turbulent media. Inset intensity patterns show the distortion effect caused by turbulent media. The mode-weights and corresponding reconstructed intensity are shown in panel (c). T−1 refers to the process of mode adjustment. (d) Measured coherence entropy in free-space (black line), through turbulent media (blue dashed line), and after mode adjustment (orange circles). Channel robustness that leverages the conservation of coherence entropy is demonstrated by sending a color-encoded image through the turbulent media, as shown in panels (e)–(g), with the correspondence between colors and values of coherence entropy shown in panel (d).