# Michelson Interferometric Methods for Full Optical Complex Convolution
## 2 Methods
It is important to highlight that achieving the proper convolution within the system is a complex process that necessitates the modulation of both amplitude and phase  
Figure 3: Experimental setup and basic performance metrics of a complex optical 4F convolution module. (**a**) A detailed photograph of the experimental setup, highlighting the zoomed-in area showcasing the fabricated amplitude mask and its precise installation within the system. This component is critical for modulating the light patterns as part of the convolution process. (**b**) A plot of the normalized intensity measurements acquired at the camera, juxtaposed with corresponding simulation data that include a confidence interval. This graph illustrates the convolution of the incoming sequence huyun111111 with the mask huyun10001], with the anticipated nine-digit result huyun11112111]. An inset image displays the actual output captured by the camera, providing a visual confirmation of the simulation accuracy and the experimental effectiveness. The theoretical shade represents the waveform of the classical convolution results of the sequence, and the math line represents the expected results of huyun11121111]. This figure shows evidence of the feasibility and accuracy of the RCCM in an experimental manner of a specific sequence. Figure 2: Simulated results demonstrating the impact of phase factors in the modulation device. (**a**) Simulated convolution incorporating both phase and amplitude factors, showcasing the comprehensive modulation capabilities and their combined effects on the resulting optical pattern. (**b**) Simulated convolution featuring only amplitude factors on the spatial light modulator (SLM), illustrating the distinct optical outcomes when phase factors are absent, thereby emphasizing the unique role of amplitude modulation in the convolution process. This comparison shows the necessity of simultaneously controlling both the phase and amplitude for the featured optical convolution module. to represent the true behavior of the featured system. The successful simulation of coherent laser light propagation and the performance of ideal lenses could offer significant insights into the behavior of real-world optical systems. Additionally, it is essential to maintain alignment accuracy at the level of the light's wavelength, \(\lambda\), along the axis of the optical path to avoid phase shifts and distortions in the interference pattern. The precision of overall misalignment, covering various alignment factors, must be within hundreds of nanometers. This encompasses the adjustment of the optical components' tip and tilt and precise positioning in the XY plane. Moreover, precise adjustment of the focal lengths of both lenses, the angle of the beam splitter, and the management of the optical path length difference between the arms of the Michelson interferometer are crucial for achieving optimal performance. Altogether, there are 10 alignment degrees of freedom that demand precision from micro- to nanometer levels to guarantee the accurate and stable functioning of the interferometry setup. During the alignment process, a collimated beam lights up both SLMs, followed by a Fourier lens converting the combined modulated beam into the frequency domain on the surface of the charge-coupled device (CCD) camera. This results in the appearance of a cross-shaped dot matrix on the camera. Ideally, with perfect alignment of the interferometer, all dots in the matrix should align perfectly. The presence of any misalignment is thus detected, enabling fine adjustments, along with the help of a Shack-Hartman sensor mounted in the optical path. The Shack-Hartman sensor can assist in cleaning up the wavefront prior to the convolution, reducing the alignment difficulties. The integration of piezo attenuators enhances this process by allowing for precise adjustments to the tip and tilt of the SLM mounts. All parts are caged in Thorlabs 30 mm caging systems to minimize possible misalignments; see Figure 3a for the experimental setup picture.