Analytical solution of the classical Rayleigh length definition, including truncation at arbitrary values

Area of Science:

• Optics and Photonics
• Optical Instrumentation Design

Background:

• The Rayleigh length is crucial for defining the depth of focus in optical systems.
• Classical definitions rely on the paraxial approximation, limiting their accuracy for high numerical apertures (NA).

Purpose of the Study:

• To develop an analytical solution for the Rayleigh length of a focused Gaussian beam applicable to any spherical truncating aperture.
• To extend the solution to include non-paraxial effects for high NA applications in optical microscopy.

Main Methods:

• Analytical solution of the diffraction integral for a focused Gaussian beam.
• Inclusion of an empirical term (Cnp) to account for the non-paraxial regime.
• Comparison with numerical calculations for validation.

Main Results:

• An exact analytical solution for Rayleigh length valid for any aperture size was obtained.
• The solution precisely matches numerical calculations in the near focal area.
• The extended solution accurately covers the entire practical range of NA (up to n*0.95) with <0.4% error in the non-paraxial limit.

Conclusions:

• The presented analytical solution offers a significant advancement for understanding and calculating Rayleigh length.
• This theoretical result is vital for optimizing light efficiency and spatial resolution in optical instrumentation.
• The model's validity across the full NA range facilitates improved optical system design.