State evolution formula and stability analysis of a paraxial optical system

Area of Science:

• Optics
• Theoretical Physics
• Mathematical Physics

Background:

• Paraxial optical systems (POS) traditionally use ray transfer matrices.
• Existing methods are often limited to linear, discrete transformations.
• Analyzing ray dynamics and stability in complex POS requires advanced formalisms.

Purpose of the Study:

• To develop a continuous dynamic equation for paraxial ray propagation and stability.
• To establish a universal framework applicable to both continuous and discontinuous optical elements.
• To derive a rigorous stability criterion for non-periodic POS.

Main Methods:

• Analysis of phase vector evolution in POS with variational refractive index.
• Derivation of the state evolution formula (SEF) as a differential equation.
• Application of SEF to determine system stability and ray trajectories.

Main Results:

• The state evolution formula (SEF) provides simultaneous phase vector transformation and ray trajectory.
• SEF extends the ray transfer matrix method to a universal differential equation.
• A rigorous criterion for the stability of continuous, non-periodic POS is established.

Conclusions:

• The SEF offers a unified and rigorous approach to analyzing paraxial ray dynamics and stability.
• This method is applicable to a wider range of optical systems than traditional matrix methods.
• The SEF serves as a valuable reference model for theoretical optical system analysis.