# Michelson Interferometric Methods for Full Optical Complex Convolution
## 3 Results and Discussion
### SLM Modulation Accuracy  
Complex convolution accuracy is reduced by inter-pixel crosstalk, an inherent property of SLMs. Ideally, the optical power at 255 and 0 pixel values should be equal,, corresponding to a full continuous \(2\pi\) modulation range. However, the inter-pixel crosstalk prevents this ideal behavior. This crosstalk occurs due to the non-local nature of the electric field that controls the liquid-crystal-on-silicon (LCOS) dipoles in each pixel of the modulatorDue to the drift of the electric field into the adjacent pixels, the experimental pixel-to-pixel modulation depth ends up being below the nominal \(2.1\pi\) range for high-density masks. The modulation range improves as the fringes widen but stays below 100%, indicating a full \(2\pi\) modulation depth has not yet been achieved.  
However, the device as a whole still performs as expected on sufficiently sparse patterns (superpixel size over 16). To assess the phase modulation efficiency of the used SLMs, we introduce a specific error rate, denoted as \(\eta\):  
\[Ae^{i\varphi_{(x,y)}}\left[\frac{e^{i\eta}\arccos\mathcal{F}_{(x,y)}+e^{-i \eta\arccos\mathcal{F}(x,y)}}{2}\right]=Ae^{i\varphi_{(x,y)}}\cos\left(\eta \arccos\mathcal{F}_{(x,y)}\right) \tag{3}\]which goes back to its original form \(A\mathcal{F}_{(x,y)}e^{i\varphi_{(x,y)}}\) when \(\eta=1\), i.e., for the ideal modulation. The parameter \(\eta\) is bounded in the huyun0, 1] range. The closer it is to 1, the better the performance of the complex convolution processor that can be expected, but when \(\eta\) is less than 0.8, the measured diffraction accuracy drops below 40%.  
Figure 4 presents several experimental results of the full RCCM. These show the convolution of different input sequences with the kernel 10001, using the Euler algorithm implemented on the SLMs, see Figure 4b for the schematic. Diffraction still has an essential impact on the results, with intensity gathering mainly in the center area of the optical path, leading to an overall error rate of 12% due to the claimed modulation depth discussed in the previous section. Also, when 0s are inserted in the input, there is always a latter slot gaining more intensity than being weighted as 0, see Figure 4c,d, which is related to the physical limitation of grating behaviors introduced by the said sharp edges when the patterns on SLMs are shifting from 2\(\pi\) to 0.  
These results hold promise that the reported algorithm is an efficient way to accomplish simultaneous phase and amplitude modulation in the Fourier domain with off-the-shelf devices capable of performing complex convolution on 2 million optical channels in parallel. Such a system is highly interesting in developing optical filters and processors for numerous applications, including optical cryptography, where neglecting phase modulation can lead to impractical bit-error rates. Therefore, we set up an optical neural network to further evaluate the featured algorithm.  
Figure 4: Optical convolution dynamics in a reconfigurable complex convolution module (RCCM) (**a**) Full experimental results of the RCCM: Convolution of the sequence \(11111\) with the kernel 10001 displayed on the spatial light modulator (SLM). This panel illustrates the primary convolution process, showcasing the direct optical output. (**b**) Adjustments with polarization-angle tuning: This modification highlights the effects of polarization-angle adjustments on the convolution output, demonstrating the influence of optical properties on the result. (**c**) Convolution result of sequence 10111 with kernel 10001: A comparison showing how minor variations in the input sequence affect the convolution outcome, providing insights into the system’s sensitivity and response. (**d**) Convolution of sequence 10011 with kernel 10001: This panel displays another variant, emphasizing how changes in the initial sequence modify the resulting convolution patterns, illustrating the dynamic capabilities of the convolution system.