Aperiodic order and Fraunhofer diffraction in one-dimensional Silver Mean gratings

We study the Fraunhofer diffraction produced by one-dimensional binary gratings based on the Silver Mean sequence, a deterministic aperiodic ordering closely related to the Pell numbers and the silver ratio. Analytical expressions for the diffraction patterns are derived within the scalar approximation, revealing a characteristic redistribution of diffracted energy and the splitting of the first diffraction orders into multiple well-defined maxima. We show that the transverse positions of these maxima are governed by simple combinations of Pell numbers, reflecting the scaling properties of the underlying aperiodic structure. The theoretical predictions are experimentally verified using a simple optical setup based on a spatial light modulator, which can be readily implemented in undergraduate teaching laboratories. These gratings provide an accessible and pedagogically valuable extension of diffraction experiments based on periodic, fractal and Fibonacci-type structures, allowing students to explore the optical consequences of deterministic aperiodic order through direct observation of their diffraction patterns.