Shape invariance and a universal form for the Gouy phase.

Area of Science:

• Optical physics
• Beam propagation
• Mathematical physics

Background:

• Coherent optical beams, including Hermite-Gaussian (HG) and Laguerre-Gaussian (LG) beams, are fundamental in various laser applications.
• Understanding beam propagation characteristics, particularly shape invariance and phase evolution, is crucial for optical system design.
• Previous research has explored properties of specific beam types, but a unified framework for shape-invariant beams was lacking.

Purpose of the Study:

• To identify all types of finite-energy coherent beams that maintain their shape during free propagation.
• To characterize the universal phase evolution (Gouy phase) of these shape-invariant beams.
• To establish the relationship between shape-invariant beams and nondiffracting beams in limiting cases.

Main Methods:

• Analysis of beam propagation using the transport-of-intensity equations.
• Application of elementary mathematical techniques.
• Utilizing the Iwasawa decomposition for a rigorous mathematical treatment.

Main Results:

• Hermite-Gaussian beams, Laguerre-Gaussian beams, and their linear combinations are identified as the sole finite-energy coherent beams exhibiting shape invariance.
• All shape-invariant beams possess a universal Gouy phase of the form c arctan(z/zR), where 'c' has quantized values.
• Two- and three-dimensional nondiffracting beams are shown to be limiting cases of shape-invariant beams as the Rayleigh distance approaches infinity.

Conclusions:

• The study provides a comprehensive classification of shape-invariant finite-energy coherent beams.
• The universal Gouy phase and quantized prefactor 'c' offer fundamental insights into beam propagation dynamics.
• The findings unify the understanding of shape-invariant and nondiffracting beams within a single theoretical framework.