PMC

Phase grating and square flattop beam profile to be extracted on SLM. k_g is the grating vector, and k_x and k_y are the normal vectors of the desired beam profile. (a) Vertical phase grating on the SLM with desired square beam profile. (b) Diagonal zigzag phase grating with desired square beam profile. (c) Vertical phase grating with desired diamond beam profile. (d) Vertical phase grating on a slanted SLM with desired square beam profile. The upper panels illustrate the SLM settings, the phase grating on the active area of the SLM, the desired beam shape (dotted red square) and the incident beam. The double-headed arrows indicate the polarisation adjusted to the direction specified for the SLM. The lower panels illustrate the phase gratings on the corresponding SLMs. The right inset explains the pixel structure on the xy-plane, and the colours indicate the phase on the pixels. Λ_D, Λ_V: phase grating period, s: pixel size, α: phase grating angle in the xy-plane and θ: SLM angle from the horizontal setting.

Desired square flattop beam profile, phase grating images and corresponding 2D-FFT images. (a) Desired square flattop beam profile on the image plane, which is extracted from the phase grating on the SLM and its 2D-FFT image. (b) Vertical phase grating image to regenerate the square beam profile and its 2D-FFT image. (c) Diagonal phase grating image to regenerate the square beam profile and its 2D-FFT image. (d) Vertical phase grating image to regenerate the diamond beam profile and its 2D-FFT image. The phase grating images are schematic and projected as they appear on the SLM, and the 2D-FFT images are projected as they appear in the Fourier plane from the SLM side. Areas of the extracted component, residual components and their overlap are shown in red, blue and violet, respectively.

In the quadrature-phase motion technique a counterphasing inducing grating (standing wave) [a, b (black line)] produces a nearly instantaneous phase-reversed counterphasing induced grating (standing wave) [a, b (red line)] in the homogeneous test field of a grating induction display. A counterphasing quadrature grating (standing wave in 90° spatial and temporal phase relative to the induced grating) [c–e (green line)] sums with the induced grating (standing wave) [c–e (red line)] to produce a rightward drifting induced-plus-quadrature grating compound (traveling wave) [c–e (yellow line)] to which the visual system is extremely sensitive. The temporal phase of the quadrature grating can be varied to measure the phase (time) lag of induction. To measure the magnitude of induction the temporal phase of the quadrature grating is held constant at 90° phase relative to the induced grating, and a second luminance grating is added to the test field. This second “canceling” grating [f, h (blue line)] possesses the same spatial and temporal frequency as the induced grating, but is 180° out of spatial phase with it. The canceling grating is added to the test field at a number of contrast levels, using the method of constant stimuli. When canceling grating contrast is less than induced grating contrast (i.e., when the induced grating is under-canceled) the compound grating possesses the spatial and temporal phase of the induced grating. It therefore combines with the quadrature grating to produce a rightward moving traveling wave just as in the case where no canceling grating is present [c–e (yellow line)]. When canceling grating contrast exceeds induced grating contrast (i.e., when the induced grating is over-canceled) the compound grating now possesses the spatial and temporal phase of the canceling grating [h (purple line)]. This compound combines with the quadrature grating to produce a leftward moving traveling wave [i–k (white line)]. When canceling grating contrast equals induced grating contrast the sum is zero (i.e., the induced grating is nulled) [f, g (purple line)]. The motion energy of the counterphasing quadrature grating which remains is left/right balanced (i.e., a standing wave) [g (green/black lines)] yielding a 50:50 proportion of left/right motion judgments in a forced-choice motion direction discrimination task.

Explanation of the quadrature-phase motion cancelation technique. (a, b) The background illustrates the appearance of a grating induction stimulus for inducing gratings at 0° and 180° temporal phase. A counterphasing inducing grating (standing wave illustrated by black line) elicits a nearly instantaneous phase reversed counterphasing induced grating (standing wave illustrated by red line) in the homogeneous test field. (c–e) A counterphasing quadrature grating (standing wave illustrated by green line) added to the test field sums with the induced grating (red line) to produce a rightward drifting induced-plus-quadrature grating compound (traveling wave illustrated by yellow line). (f) A canceling grating (blue line) added to the test field in the same temporal phase as the induced grating (red line) but in 180° spatial phase is used to null the induced grating (purple line). (g) When the induced grating is nulled by the canceling grating the quad grating which remains is motion-energy balanced and causes no systematic unidirectional motion. (h) When the induced grating is over-canceled the induced grating plus canceling grating compound (standing wave illustrated by purple line) has the spatial and temporal phase of the canceling grating (blue line). (i–k) The over-canceled induced-plus-canceling grating compound sums with the quadrature grating to produce a leftward drifting compound (traveling wave illustrated by white line).