Fig. 1. Schematic of an optical system using a DOE

Fig. 2. Flow charts of GS algorithm (left) and improved GS algorithm (right)

Fig. 3. Schematic of amplitude constraint of the output plane, where the region A of the output plane is divided into 9 large square regions and other regions [amplitude operation (Ⅰ)] and the large square region consists of small square region [amplitude operation (Ⅱ)] and square box 1, square box 2 and square box 3 [amplitude operation (Ⅲ)] from small to large. (a) Amplitude of the output plane; (b) magnified image of the region A in Fig. 3(a)

Fig. 4. Results of simulation analysis of normal diffraction limited spot array and sub-diffraction spot array (SSA). (a)(b) Intensity distributions of normal diffraction limited spot arrays and SSA; (c) fitted normalized intensity profiles along the dashed line across a single spot in the central of the normal diffraction limited spot arrays and the SSA; (d) fitted normalized intensity profiles along the dashed line across the three points

Fig. 5. Schematic of 2P-sMSIM system

Fig. 6. Imaging results of normal diffraction limited spot array and SSA. (a)(b) Images of normal diffraction limited spot array and SSA; (c) intensity profiles along the dotted lines in Fig. 6(a) and Fig. 6(b)

Fig. 7. 2P-sMSIM imaging of microtubules in a BS-C-1 cell. (a) Wide-field image obtained with normal diffraction limited spot array excitation (obtain with overlapping all spot arrays); (b) wide-field image obtained with SSA excitation (obtain with overlapping all spot arrays); (c) 2P-MSIM imaging; (d) 2P-sMSIM imaging; (e) intensity profiles along the dotted lines in Fig. 7(c) and Fig. 7(d)

Fig. 8. 2P-sMSIM imaging of mitochondria. (a) Wide-field image obtained by using normal diffraction limited spot excitation; (b) wide-field image obtained by using SSA excitation; (c) 2P-MSIM imaging; (d) 2P-sMSIM imaging; (e)(f) magnified view of the rectangle region boxed in Fig. 8(c) and Fig. 8(d)