Focal shifts in diffracted converging electromagnetic waves. I. Kirchhoff theory

Area of Science:

• Optics and electromagnetism
• Diffraction theory
• Computational physics

Background:

• Kirchhoff's diffraction theory provides a foundational model for wave propagation.
• Standard assumptions in diffraction theory can limit accuracy, particularly near edges.
• Understanding axial energy distribution and focal shift is crucial for optical system design.

Purpose of the Study:

• To develop a more accurate vector formulation of Kirchhoff diffraction theory.
• To analyze the impact of edge diffraction integrals on axial energy density.
• To investigate the focal shift in optical systems considering numerical aperture and Fresnel number.

Main Methods:

• Vector formulation of Kirchhoff diffraction theory.
• Inclusion of a line integral around the aperture edge.
• Numerical examination of axial field distribution.
• Analysis of focal shift for aplanatic systems and parabolic mirrors.

Main Results:

• Expressions for total energy density distribution along the axis were derived.
• The numerical consequence of neglecting the edge integral was quantified.
• Focal shift was evaluated for systems with arbitrary numerical aperture and finite Fresnel number.
• Combined effects of Fresnel number and numerical aperture on focal shift were determined.

Conclusions:

• The edge integral contributes significantly to the axial field and focal shift.
• Accurate modeling requires considering edge effects in diffraction analysis.
• The study provides a more rigorous framework for understanding diffraction in optical systems.