Fig. 1. A block diagram representation of the proposed discrete fractional convolution operation defined in Eq. (1), where the DFrFT operator is described by the unitary operator Eq. (2). The symbol ⊙ represents the pointwise or Hadamard product.

Fig. 2. Integrated photonic circuit sketch for the proposed DFrFT-based convolution accelerator. The architecture comprises two DFrFT lattices of normalized lengths π/2 and 3π/2, rendering the direct and inverse DFrFT of order α=π/2, respectively. The intermediate pointwise multiplication is performed through an array of programmable phase shifters. These are electrically tuned by the phase sets {eiθn} and {eiϕn} so that this layer performs K(α)≔Fα[k].

Fig. 3. (a) Convolution of the eigenmodes u(20) (left column) and u(10) (right column) with the convolution kernel k(δ2,0) (upper row) and k(δ,0) (lower row) for j=20 (N=41). (b) Convolution of the eigenmodes u(100) (left column) and u(90) (right column) with the convolution kernel k(δ2,0) (upper row) and k(δ,0) (lower row) for j=100 (N=201).

Fig. 4. Continuous interpolation of the distance function dy1,y2=∥y1−y2∥ [Eq. (3)] between the input eigenmodes y1=u(n) and the corresponding normalized convolution operation using the kernel y2=k(δ2,0). The distance has been plotted as a function of the eigenmode number n∈{−j,…,0,…,j} for j=20,50,100.

Fig. 5. (a) Convolution of the eigenmodes u(15) (upper row) and u(18) (lower row) with the kernel k(δ2,r) for r=5 (yellow), r=10 (green), and r=15 (red) for j=20. (b) Convolution of the eigenmodes u(95) and u(98) with the kernel k(δ2,r) for r=25 (yellow), r=50 (green), and r=75 (red) for j=100.

Fig. 6. Convolution norm ‖(f*π/2k)q‖ using the u(j) filter and a noisy signal f (lower inset) composed of a rectangular window plus Gaussian noise for j=100. The upper-left and upper-right insets depict the even and odd pooling, respectively, associated with the convolution output.

Fig. 7. Input signal (black-shaded area) and the corresponding convolution norm using the filter u(j−1) (lower row) and double-box (Canny edge) filter k(c) (upper row) for a waveguide array with j=100.

Fig. 8. Edge detection scheme using the noisy rectangular signal of Fig. 6 and the convolution kernel u(j−1) (lower row) and the double-box filter k(c) (upper row) with j=100. In both cases, the upper-left and upper-right insets depict the even and odd pooling of the convolution output, respectively.

Fig. 9. (a) Propagation of the electric field modulus |E(x,y)z^| throughout the convolution architecture. (b) Input and output field amplitudes of the simulated results, where the gray-shaded area denotes the location of the waveguide core. (c) Theoretical predictions from coupled-mode theory.

Fig. 10. Mean (line) and standard deviation (shaded area) of the relative percent error [E(δ)×100%] for a set of 100 randomly generated perturbed DFrFT matrices [G(π/2;δ)] per value of δ. The insets depict |G(π;δ)| (matrix modulus) of a random sample out of the set of 100 random matrices for δ=0.05,0.1,0.3.

Fig. 11. Distance function (dy1,y2(δ)) between the unperturbed (δ=0) and perturbed (δ≠0) convolution operations using the (a) u(j−1) and (b) double-box filters. The plot is presented as a function of δ and the lattice error E(δ)×100%. The mean (line) and standard deviation (shaded area) are computed by generating a set of 100 random perturbed DFrFT matrices [G(π/2;δ)] per value of δ. The insets depict the output of the perturbed filtering using the even (yellow) and odd (red) pooling operation for δ=0.05,0.1,0.2.