Exactly unitary discrete representations of the metaplectic transform for linear-time algorithms

Area of Science:

• Optics and Photonics
• Mathematical Physics

Background:

• The metaplectic transform (MT), also known as the linear canonical transform, is crucial in optics for analyzing light beam transformations.
• Existing approximations of the near-identity MT (NIMT) lack unitarity, causing numerical instability in geometrical-optics modeling of caustics.

Purpose of the Study:

• To develop a discrete metaplectic transform (MT) that is exactly unitary.
• To create a discrete near-identity MT (NIMT) approximation that is also unitary and computationally efficient.
• To ensure the discrete NIMT converges to the discrete MT for broader applicability.

Main Methods:

• Development of an exactly unitary discrete metaplectic transform (MT).
• Approximation of the discrete MT to create a unitary discrete near-identity MT (NIMT).
• Algorithmic implementation for linear-time computation and iterative convergence proof.

Main Results:

• A novel, exactly unitary discrete metaplectic transform (MT) was successfully developed.
• A unitary discrete near-identity MT (NIMT) was derived, offering linear-time computation and improved numerical stability.
• The discrete NIMT was proven to converge to the discrete MT upon iteration.

Conclusions:

• The new discrete MT and unitary NIMT algorithms provide a stable and efficient tool for optical system analysis.
• These advancements enhance geometrical-optics modeling, particularly for caustics, by overcoming previous numerical limitations.
• The developed algorithms broaden the applicability of NIMT for computing general MTs, not just near-identity cases.