## 2 Theoretical Considerations  
The monochromatic retroreflection mechanism from the parallel-superimposed bi-grating structure relies on the coordinated interaction between the two individual gratings to in-couple and out-couple incident light into the SPR mode. As described in previous studies [27], a grating couples incident light into a plasmon mode when the phasing matching condition is satisfied. The reverse principle can also occur, in which a grating can out-couple the SPR light, thus directionally redirecting the SPR light back toward the source. Combining both SPR in-coupling and out-coupling using a bi-grating by carefully matching the grating pitches can permit efficient retroreflection at a specific narrowband wavelength and angles of incidence, as depicted in Figure 1a. The underlying mechanism is explained through the dispersion curve relationship, which describes the interaction between the frequency of light (\(\omega\)) and the wavevector of the surface plasmon resonance (\(k_{SPR}\)) at the metal-dielectric interface. The dispersion relation for SPR using a grating can be expressed as  
\[k_{SPR}=\frac{\omega}{c}\sin\theta+K \tag{1}\]where \(\omega=\frac{2\pi}{\lambda}\), \(\lambda\) is the wavelength of the incident light, \(c\) is the speed of light in vacuum, \(\theta\) is the angle of incidence of the light beam relative to the grating surface, and \(K=\frac{2\pi}{\Lambda}\) is the reciprocal vector of the grating, and \(\Lambda\) is the grating pitch (period). This equation describes how the incident light's wavevector must equate the surface plasmon polariton (SPP) wavevector to excite an SPR mode. The periodic structure of the grating enables this phase-matching condition where the incident light's wavevector is matched to that of the SPR mode. Figure 1b contains the experimental SPR wavelength response data, calculated using Equation (1), to obtain the dispersion relationship for an array of single sinusoidal gold-coated surface relief gratings. This figure shows how the SPR wavevector is excited at varying wavelengths (the black dots in the figure) based on the changing pitch of each individual grating. In the parallel-superimposed bi-grating system, the two gratings work in tandem to achieve SPR-enhanced retroreflection, as depicted in Figure 1c, at an arbitrary angle \(\theta=+6^{\circ}\). The first grating (pitch of \(K_{1}\)) couples the incoming light into the SPR mode along the gold-air interface towards the forward-propagating direction. The second grating (pitch of \(K_{2}\)) is designed to have a pitch that matches the phase of the SPR to out-couple the SPR, thus redirecting the in-plane light back in the reverse direction at an angle \(\theta=-6^{\circ}\), as depicted by the dotted line in Figure 1c. This process ensures that the light is efficiently reflected back along its original path, fulfilling the retroreflective function. The in-coupling / out-coupling mechanism of SPR allows the device to selectively retroreflect light at specific wavelengths and angles, offering a precise and tunable optical  
Figure 1: (**a**) Schematic representation of two separate gratings, which, when superimposed, form a parallel-superimposed bi-grating structure capable of retroreflection. The diagram illustrates the retroreflection process, including the incident wavevector, diffraction orders, excitation of surface plasmon resonance (SPR), and the back-coupled retroreflected light. (**b**) The SPR dispersion curve for a sinusoidal gold-coated grating plotted with the light curve. (**c**) Dispersion curve illustrating the in-coupling and out-coupling of SPR in the parallel-superimposed bi-grating structure at an arbitrary angle \(\theta=\pm 6^{\circ}\). The dashed line indicates the in-coupling of incident light into SPR by the first grating (pitch \(K_{1}\)) at \(\theta=+6^{\circ}\), while the dotted line shows the out-coupling of SPR by the second grating (pitch \(K_{2}\)), resulting in retroreflection at \(\theta=-6^{\circ}\). response using the same device.