Vectorial ray-based diffraction integral

Abstract

Laser interferometry, as applied in cutting-edge length and displacement metrology, requires detailed analysis of systematic effects due to diffraction, which may affect the measurement uncertainty. When the measurements aim at subnanometer accuracy levels, it is possible that the description of interferometer operation by paraxial and scalar approximations is not sufficient. Therefore, in this paper, we place emphasis on models based on nonparaxial vector beams. We address this challenge by proposing a method that uses the Huygens integral to propagate the electromagnetic fields and ray tracing to achieve numerical computability. Toy models are used to test the method's accuracy. Finally, we recalculate the diffraction correction for an interferometer, which was recently investigated by paraxial methods.

Diffraction effects in length measurements by laser interferometry

“…In high-accuracy dimensional measurements by laser interferometry, one of the largest corrections required is due to diffraction. Examples are the measurements of Si lattice-parameter by combined X-ray and optical interferometry [1][2][3][4], of gravity by free-fall gravimeters [5][6][7], and of diameters by Fizeau interferometers [8][9][10][11][12][13]. At the 10 −9 level of relative accuracy it is not possible to trace back wavelength to frequency via the plane-wave dispersion equation.…”

Ray-based method for simulating cascaded diffraction in high-numerical-aperture systems

“…We compare results obtained using vectorial HFPI with those of two other simulation methods. The first method is the vectorial ray-based diffraction integral (VRBDI) of Andreas et al [2] for which the authors have published their MATLAB code online [1]. The second reference method is beam synthesis propagation (BSP) in the commercial software package CODE V [5].…”

“…To simulate optical system comprised of multiple optical media we therefore choose a different approach based on a physical interpretation of these diffraction integrals. Equations (1,2) are integrations over the following two quantities…”

“…Like scalar HFPI, vectorial HFPI is robust but computationally intensive. It bears some conceptual resemblance to the vectorial ray-based diffraction integral (VRBDI) introduced by Andreas et al [2], but its implementation and intended domain of application differ. While VRBDI was developed to simulate the propagation of a diffracting beam (i.e., a laser beam) through a non-diffracting optical system for interferometry, HFPI is developed for the simulation of multiple diffraction in optical systems like the before mentioned HMDs.…”

Simulating multiple diffraction in imaging systems using a path integration method

“…Andreas et al use the Huygens-Fresnel principle to propagate the vectorial field through a three-dimensional system. Their method includes diffraction due to the propagation of the input field but considers nondiffracting optical systems [12].…”