Fig. 1. Schematic illustration of Fermat’s spiral. (a), (c) Geometric pattern. (b), (d) Phase distribution. (a1), (b1) Transformation phase and (a2), (b2) correction phase with n=8/5. (c1), (d1) Transformation phase and (c2), (d2) correction phase with n=4/5.

Fig. 2. Numerical simulations of Fermat’s spiral transformation on vortex beams in the case of integer multiplication and division. (a) Input LG beams with ℓ=−6,−4,+4,+6. (b) Output beams while n=2. (c) Corresponding intensity and phase of (b). (d) Output beams while n=1/2. (e) Corresponding intensity and phase of (d).

Fig. 3. Numerical simulations of Fermat’s spiral transformation on vortex beams in the case of fraction multiplication and division. (a) Input LG beams with ℓ=−3,+3,−6,+6. (b1), (b2) Output beams while n=5/3. (b3), (b4) Output beams while n=2/3. (c) Corresponding intensity and phase of (b).

Fig. 4. Phase with different transformation parameters n that are applied to Figs. 2 and 3. (a1)–(d1) Transformation phase. (a2)–(d2) Correction phase. (a1), (a2) n=2. (b1), (b2) n=1/2. (c1), (c2) n=5/3. (d1), (d2) n=2/3.

Fig. 5. Power weight of the output OAM modes. (a) Multiplication with n=2. (b) Division with n=1/2.

Fig. 6. Optical description of integer multiplication with n=2. (a1)–(a4) Intensity of input LG beams with ℓ=−3,+2,+3,+4. (b1)–(b4) Corresponding interference patterns of (a). (c1)–(c4) Intensity of the output beams with ℓ=−6,+4,+6,+8. (d1)–(d4) Corresponding interference patterns of (c). All images are on the same size scale.

Fig. 7. Optical description of integer division with n=1/2. (a1)–(a4) Intensity of input LG beams with ℓ=−6,+4,+6,+8. (b1)–(b4) Corresponding interference patterns of (a). (c1)–(c4) Intensity of the output beams with ℓ=−3,+2,+3,+4. (d1)–(d4) Corresponding interference patterns of (c). All images are on the same size scale.

Fig. 8. Optical description of fraction multiplication with n=3/2 and division with n=3/4. (a1)–(a4) Intensity of input LG beams with ℓ=−4,−6,+8,+12. (b1)–(b4) Corresponding interference patterns of (a). (c1)–(c4) Intensity of the output beams with ℓ=−6,−9,+6,+9. (d1)–(d4) Corresponding interference patterns of (c). All images are on the same size scale.

Fig. 9. Schematic of the experimental setup. HP, half-wave plate; Pol.1–Pol.3, polarizer; ×10 objective, objective lens (×10, NA=0.25); L1, 60 mm lens; SLM1, transmissive spatial light modulator; BS, beam splitter; SLM2, reflective phase-only spatial light modulator; L2 and L3, 150 mm lens; CMOS, complementary metal oxide semiconductor camera.

Fig. 10. Evolution of input LG beams carrying ℓ=4 in the case of integer multiplication and division. (a) Simulation results of the transverse intensity at different distances while n=2. (b) Corresponding experiment results of (a). (c) Simulation results of the transverse intensity at different distances while n=1/2. (d) Corresponding experiment results of (c).

Fig. 11. Evolution of input LG beams carrying ℓ=5 in the case of fraction multiplication and division. (a) Simulation results of the transverse intensity at different distances while n=8/5. (b) Corresponding experiment results of (a). (c) Simulation results of the transverse intensity at different distances while n=4/5. (d) Corresponding experiment results of (c).