Exact mirror equation via Berry's caustic touching theorem: plane and spherical mirrors

Area of Science:

• Optics
• Geometrical Optics
• Classical Physics

Background:

• Understanding image formation by mirrors is crucial in geometrical optics.
• Existing models often rely on approximations, limiting their applicability.
• The behavior of reflected light rays near caustics requires precise mathematical description.

Purpose of the Study:

• To derive an exact set of equations for image position in mirrors within the geometrical optics approximation.
• To analyze the regions of reflected light ray reception for observers.
• To demonstrate the connection between exact equations and the paraxial approximation.

Main Methods:

• Utilizing the geometrical optics approximation for light propagation.
• Applying the caustic touching theorem.
• Deriving and analyzing exact equations for image formation.
• Investigating specific cases for plane and spherical mirrors.

Main Results:

• An exact set of equations for image position was obtained.
• The number of reflected light circles (zero, one, two, or three) received by an observer was determined for different mirror types.
• The derived exact equations reduce to the well-known mirror equation under the paraxial approximation.

Conclusions:

• The study provides a rigorous framework for image formation in mirrors beyond paraxial assumptions.
• The analysis of reflected light ray regions offers insights into optical phenomena near caustics.
• The work bridges exact geometrical optics with the commonly used paraxial approximation.