Fig. 1. (a) Alignment-free fractal multipoint interferometer. Laser, He–Ne laser with 633 nm wavelength; L1, 50 mm lens; L2, 500 mm lens; SLM, phase-only spatial light modulator; L3, 300 mm lens; P, pinhole; L4, 300 mm lens; DMD, digital micromirror device; L5, 250 mm lens; CCD, charge-coupled device. (b) proposed FMM; (c) example of the far-field intensity patterns.

Fig. 2. Schematic diagram of DenseNet-121. CONV, convolution layer; MP, max pooling layer; DB, dense block; GMP, global max pooling layer; FC, fully connected layer.

Fig. 3. Examples of the experimental diffraction intensity patterns for LG eigenstates with topological charge ℓ∈{0,1,±2,5} and different FMM displacements Δx=Δy∈{0,0.15,0.25}  mm. In addition, all the diffraction patterns are obtained with stochastic disturbances of the other two parameters: (i) beam waist size ω∈[0.45,0.55]  mm; (ii) initial phase of OAM states φ0∈[0,2π]. Insets in (e1)–(e3) show the detailed profiles of the recorded intensity patterns.

Fig. 4. Confusion matrix for the recognition of misaligned LG eigenstates ℓ∈{−5,−4,⋯,5} and p=0, with the curves of accuracy and loss as functions of epochs on the top right.

Fig. 5. Schematic diagram of a Bloch sphere constructed with |ℓ=±1⟩ bases. (a) An arbitrary state |ψ⟩ on the sphere and two adjacent superposition states ρA and ρB; (b) sphere is divided into 80,000 points corresponding to 80,000 states; the red box schematically indicates the position distribution of the nine selected superposition states for the experiments.

Fig. 6. Experimental results of hyperfine LG superposition states. (a)–(c) Examples of the recorded diffraction intensity patterns for OAM superposition states under different misaligned configurations. The collection of each diffraction pattern in the figure is carried out with stochastic disturbances of the other two parameters: (i) beam waist size ω∈[0.45,0.55]  mm; (ii) initial phase of OAM states φ0∈[0,2π]. To show the image clearly, we reduce the contrast of the image by setting the values of I=0.3×max{I}, where I>0.3×max{I}; (d) confusion matrix for superposition states from Mode 1 to Mode 9 with the curves of accuracy and loss as functions of epochs on the top right.

Fig. 7. Confusion matrix of LG modes with p=1. (a) Confusion matrix for LG eigenstates of ℓ∈{−2,−1,⋯,2} and p=1; (b) confusion matrix for superposition states from Mode 4 to Mode 8. Mode 4 to Mode 8 here represent the superposition states with p=1 at the same positions on the Bloch sphere in the case of p=0.

Table 1. Nine Superposition States Combined by θ and ϕ