Fig. 1. Interference between diffracted vortex beam and diffracted plane wave beam through a phase aperture in common-path interferometry. (a) Case I: the inner circle zone (≤r1) is covered with the helical phase-modulated profile exp(ilθ) with the topological charge l=3. The ring zone (≥r1,≤r2) is without phase modulation and outputs a Gaussian beam. (b) Case II: the circle zone (≤r1) is without phase modulation. The ring zone (≥r1,≤r2) is covered with the helical phase-modulated profile exp(ilθ) with the topological charge l=3.

Fig. 2. Experimental setup of common-path interferometry with a phase aperture. λ/2, half-wavelength plate; L1, L2, lens; P, pinhole; BS, beam splitter; SLM, spatial light modulator; CCD, charge-coupled device. r1 is the inner radius of the phase aperture, r2 is the outer radius of the phase aperture indicated by red dashed line.

Fig. 3. (a) Circular phase-aperture element with a helical phase-modulated profile in the inner circle zone. (b) Numerically simulated light intensity distribution modulated by (a). (c) Experimentally recorded intensity distribution modulated by (a). (d) Circular phase-aperture element with a helical phase-modulated profile in the ring zone. (e) Numerically simulated light intensity distribution modulated by (d). (f) Experimentally recorded intensity distribution modulated by (d). The red dashed circle indicates the outer radius r2 of the phase aperture.

Fig. 4. (a) Triangular phase-aperture element with a helical phase-modulated profile in the inner zone. The inner equilateral triangle has the side length d. (b) Numerically simulated intensity distribution modulated by (a). (c) Experimentally recorded intensity distribution modulated by (a). (d) Circular phase-aperture element with a helical phase-modulated profile in the zone between the inner triangle and outer ring. (e) Numerically simulated intensity distribution modulated by (d). (f) Experimentally recorded intensity distribution modulated by (d). The red-dashed circle indicates the outer radius r2 of the phase aperture.

Fig. 5. (a), (d), (g), (j) Triangular phase-aperture element with a helical phase-modulated profile in the inner zone. The triangular phase aperture has a different side length, d. (b), (e), (h), (k) Numerically simulated intensity distribution and (c), (f), (i), (l) experimental intensity distribution in the far-field. The red dashed circle indicates the outer radius r2 of the phase aperture.

Fig. 6. (a), (d), (g), (j) Triangular phase-aperture element with a helical phase-modulated profile inside. The orientation of the triangular phase aperture tilts with different angle θ; the angle θ=10deg, 30 deg, 60 deg. (b), (e), (h), (k) Numerically simulated intensity distribution and (c), (f), (i), (l) experimentally generated intensity pattern. *The white dots indicate the changed direction of the interference fringes. The red dashed circle indicates the outer radius r2 of the phase aperture.

Fig. 7. (a) Triangular phase-aperture element with a helical phase-modulated profile inside. The outer zone of the phase aperture has 0 phase shift. (b) Numerically simulated intensity distribution. (c) Experimental recording. (d) The outer zone of the phase aperture has π phase shift. (e) Numerically simulated intensity distribution. (f) Experimental recording. The white dots indicate the changed direction of the interference fringes. The red dashed circle indicates the outer radius r2 of the phase aperture.

Fig. 8. (a), (b) Triangular phase aperture with a helical phase-modulated profile inside (l=4 and l=8). (c)–(e) Rectangular phase aperture with a helical phase-modulated profile inside (l=4,8,16). The red dashed circle indicates the outer radius r2 of the phase aperture. The first column illustrates the phase modulation profile; the second column shows the calculated light intensity distribution, and the third column shows the experimental results.