Tailoring on-axis spectral density with circularly coherent light beams.

Area of Science:

• Optics and Photonics
• Electromagnetism
• Wave Propagation

Background:

• Understanding the spectral properties of light beams is crucial in various optical applications.
• Characterizing the cross-spectral density (CSD) provides insights into the spatial and spectral coherence of optical fields.
• Previous methods for determining CSD were often complex or limited in scope.

Purpose of the Study:

• To derive a closed-form expression for the on-axis CSD of a beam from a stationary source with circular coherence and Gaussian spectral density.
• To establish relationships between the on-axis CSD and fundamental source properties like degree of coherence and pseudo-mode weighting functions.
• To demonstrate the utility of these relationships for designing and controlling the spectral characteristics of optical beams.

Main Methods:

• Analytical derivation of the on-axis CSD using mathematical physics principles.
• Application of integral transforms, specifically the Laplace and Hilbert transforms.
• Numerical simulations to validate the derived formulas and explore spectral tailoring possibilities.

Main Results:

• A closed-form equation for the on-axis CSD was successfully obtained.
• The on-axis CSD was shown to be directly related to the Laplace transform of the source's degree of coherence.
• An alternative expression involving the Hilbert transform of the pseudo-mode weighting function was also derived.
• Numerical examples illustrated the effective control over on-axis spectral density achievable through these relationships.

Conclusions:

• The derived closed-form solution simplifies the analysis of on-axis CSD for specific Gaussian beams.
• The established mathematical connections offer a powerful tool for engineers and physicists to tailor spectral properties.
• This work facilitates the design of optical systems with desired spectral characteristics by manipulating source coherence and spectral properties.