# Michelson Interferometric Methods for Full Optical Complex Convolution
## 2 Methods  
In setups that require controllable amplitude and phase modulation, it is advantageous to switch between the two regimes on demand. Additionally, for several applications, such as heterogeneous computing and optical neural networks, optical image processing, and holography, achieving a full optical convolution potentially provides higher performance and more efficient information harvesting due to information pre-processing in the domain native to the data origin. To realize a simultaneous phase and amplitude convolution in the Fourier domain with phase-only SLMs, we employ a Michelson interferometer, as shown in Figure 1a,b. An incoming table-top laser beam \(Ae^{(i\phi}0)\) propagates through the Michelson setup, where it is split by a 50/50 beam splitter into two equivalent arms; see Figure 1a. Each component picks up a mix of pre-calculated amplitude and phase modulation from the two SLMs (Holoeye Pluto-2 VIS-096), see insert in Figure 1b, resulting in the wavefront becoming  
\[Ae^{i\varphi(x,y)}\left[\frac{e^{i}\arccos\mathcal{F}(x,y)+e^{- \arccos\mathcal{F}(x,y)}}{2}\right]=A\mathcal{F}_{(x,y)}e^{i\varphi(x,y)} \tag{1}\]  
where \(A\) denotes the constant amplitude of the incoming laser beam; and \(\varphi\) and \(\arccos\)\(F\left(x,y\right)\) are the phase and the amplitude modulations to be imprinted by the designed setup, respectively. The phase masks are thus  
\[M_{(i,j)}^{(1,2)}=\varphi_{(i,j)}^{(1,2)}\pm\arccos\mathcal{F}_{(i,j)}^{(1,2)} \tag{2}\]  
Here, indices _[i, j] represent_ the pixels of a corresponding active region on each SLM, and (1, 2) correspond to the first SLM (SLM1) and second SLM (SLM2). To accomplish full optical convolution, SLM1 modulates the amplitude of an incoming table-top beam. After that, within the Fourier domain of a 4F system, transferred passively by a piece of lens, SLM2 and SLM3 further modulate the combined complex-valued phase and amplitude factors by projecting pre-calculated Fourier-transformed masks representing the 2-D distribution of the phase and amplitude combination, denoted as M. For a view of the experimental setup, refer to Figure 1b. Three computer-generated SLM patterns representing different convolution pairs are shown sequentially in this figure, and their results, acquired by a camera, are shown in the same sequence above the camera icon. Figure 2a illustrates the simulation of the featured algorithm, while Figure 3a displays an image of the actual implementation. To ensure precision and minimize possible misalignments, all components are integrated within a robust Thorlabs 30 mm caging system, providing a stable and reliable framework for the optical configuration. Figure 1: Schematic and working principle of a complex optical 4F convolution module. (**a**) A detailed schematic representation illustrating the complex convolution process based on [PERSON]’s formula, emphasizing the mathematical underpinnings and optical path integration. (**b**) An expanded diagram of the experimental setup for executing full optical convolution. SLM1 modulated the illuminating beam’s amplitude, and SLM2 and 3, as a whole, sitting in the Fourier domain, modulated the beam with phase and amplitude controlled simultaneously; see Equation (1). A comprehensive simulation was performed within a Python environment to replicate the propagation of coherent laser light and to examine the functionality of ideal lenses, utilizing Fresnel diffraction theory, accomplished with the Lightpipes simulator [21]. All SLM masks were pre-calculated through mathematical optical propagation, following the Fresnel diffraction formula, trigonometric functions, and [PERSON]'s law to simulate the approximate Fourier transform in an experimental environment. One of the critical stages of the simulation is the normalization of masks within the range of \(0\) to \(2\pi\). This normalization ensures that the correct phase-space representation is achieved, maintaining the integrity of the optical signals as they are processed through the system.