Abstract
Abstract The physical meaning and essence of Fresnel numbers are discussed, and two definitions of these numbers for off-axis optical systems are proposed. The universal Fresnel number is found to be $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}N=(a^{2}/\lambda z ) \ast C_{1} +C_{2} $ . The Rayleigh–Sommerfeld nonparaxial diffraction formula states that a simple analytical formula for the nonparaxial intensity distribution after a circular aperture can be obtained. Theoretical derivations and numerical calculations reveal that the first correction factor $C_{1} $ is equal to $\cos \theta $ and the second factor $C_{2} $ is a function of the incident wavefront and the shape of the diffractive aperture. Finally, some diffraction phenomena in off-axis optical systems are explained by the off-axis Fresnel number.
Highlights
Diffraction fields can be exactly solved by the Fresnel diffraction integral, but the calculation is highly complicated
The correction factor C2 = a2/λRc is identical to the factor under normal incidence, indicating that the factor is not related to the incident angle, but is determined by the curvature radius of the incident wavefront; this result proves the correctness of the off-axis Fresnel number
C2 is independent of the incident angle; this value is a function of the aperture shape
Summary
Diffraction fields can be exactly solved by the Fresnel diffraction integral, but the calculation is highly complicated. The essence of the Fresnel number is the variation of the optical path in the propagation; the physical meaning is the number of Fresnel half-wave zones included in the diffractive aperture[4]. The Fresnel number can be expanded to the off-axis point; the number of Fresnel half-wave zones for this point can be calculated to elucidate the properties of radially diffracted fields. The physical meaning and application of complex Fresnel numbers in Gaussian beams diffracted by hard apertures have been studied[10]. Oblique incidence[11] and tilted optical elements[12] are often used in practice; a universal Fresnel number to explain the diffraction phenomena in an off-axis optical system is necessary. In this paper an expression for the off-axis Fresnel number is provided through theoretical derivations and numerical calculations
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