Abstract
Keywords
1. Introduction
Diffraction fields can be exactly solved by the Fresnel diffraction integral, but the calculation is highly complicated. The Fresnel number allows a qualitative or semi-quantitative analysis of the diffraction field, thereby providing a clear physical picture and intuitive method. Some examples are the connections between the Fresnel number and the focal shift[
If a plane wave is normally incident upon a circular aperture, the standard Fresnel number is defined as[
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[
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. For example, the locations of the minimal and secondary maximal intensities in the radially diffracted field of a grating can be obtained by the half-wave zone method[
Light beams fall under normal or oblique incidences in optical systems, and their Fresnel number of normal incidence is given in Refs. [
Figure 1.Diagram representing off-axis beams.
Figure 2.Beam propagation in the (a) meridian plane and (b) the sagittal surface.
2. Definition of the off-axis Fresnel number
The Fresnel number is defined in two ways. First, the number physically represents the optical path difference between a wavelet from the aperture edge to the observation point and a wavelet from the aperture center to the observation point; the path difference is then divided by [
By taking the direction of the center wavelet as the auxiliary axis , the focal point is defined as the intersection of the auxiliary axis and the observation screen (Figure
3. Resolution of the expression for the off-axis Fresnel number
In the case of a plane wave, the off-axis Fresnel number is obtained through theoretical derivations and numerical calculations. The scope of application is discussed by comparison with the result obtained through numerical calculations.
3.1. Theoretical derivation
The coordinate system is chosen so that the incident plane wave is perpendicular to , subtends an angle of with the axis in the counterclockwise direction and forms an angle of with the optical axis . Taylor series expansion is carried out around if is taken as the auxiliary axis. The meridian plane comprises axes and ; the sagittal surface consists of axes and . The optical path differences are calculated separately in these two surfaces (Figure
3.1.1. Meridian plane
As shown in Figure
(1) The optical path of the upper wavelet (DAC).
(2) The optical path of the lower wavelet (BC).
(3) The optical path of the center wavelet (EC).
It can be seen that the optical paths on the upper and lower edges are equal ; the difference between optical paths on the edge and on the center is given as
3.1.2. Sagittal surface
(1) The optical paths of the wavelets from the upper and lower edges are equal (as shown in Figure
(2) The optical path of the wavelet from the center.
(3) The optical path difference.
The derivation implies that the optical path differences in the meridian plane and the sagittal surface are equal and are given as ; the off-axis Fresnel number can be expressed as
Figure 3.The diffraction patterns obtained by the R–S (upper panel) and analytical (lower panel) formulas.
3.2. Numerical calculation
The Rayleigh–Sommerfeld (R–S) nonparaxial diffraction integral is adopted to handle beams under oblique incidence; some approximations are introduced to avoid complicated calculations. A simple analytical formula for the nonparaxial intensity distribution is derived to reduce the computational complexity.
3.2.1. Nonparaxial intensity diffraction behind a circular aperture
The obliquely incident beam is no longer paraxial; hence, the R–S formula is used to calculate the distribution of the diffracted field[
Although the R–S formula can accurately calculate the nonparaxial scalar diffraction field and yield an accurate off-axis Fresnel number, it is difficult to obtain a universal expression for this number because of the complexity of the mathematical and numerical calculations. Thus, it is necessary to adopt an effective approximation to obtain a concise expression for the off-axis Fresnel number. is expanded into a Taylor series around [
The circular function is expressed as a series expansion with a complex Gaussian function[
Using the expression into Equation (
3.2.2. In silico simulations
(1) Comparison with the diffraction pattern.
Taking , , , , the diffracted patterns (Figure
Both patterns are roughly similar; however, a difference still exists because the analytical formula is derived under approximation conditions. The diffraction pattern obtained by the R–S formula is not circular; the radial intensity distributions are different in the horizontal and vertical directions. The diffraction pattern obtained using the analytical formula exhibits the same intensity distribution in both directions; the patterns are coincident with those obtained in Section
(2) Off-axis Fresnel number derived through numerical calculations.
From the definition of the equivalent off-axis Fresnel number, and curves are acquired from analytical calculations. Figure
Figure 4. (a) and (b) curves.
The simple analytical formula is used to calculate the off-axis Fresnel number. Based on the standard Fresnel number, the off-axis Fresnel number is expressed as , where is the correction factor. Through curve fitting, the correction factors under different incident angles are obtained from the simulation. The solid line in Figure
The off-axis Fresnel number can be expressed as
The same expression is obtained in Section
Figure 5.Curve of .
3.3. Scope of application
The off-axis Fresnel number is established under certain approximations. Hence, the ratio of the propagation distance and radius has to meet specific conditions. The accuracy of the Fresnel number obtained by the R–S formula allows comparison of the curves derived from the R–S and analytic formulas, which gives the scope of application of the expression. Figure
Figure 6. curves under different incident angles.
The curves agree with each other well for ; the expression derived in this paper is established for a relatively fine beam. When the propagation distance is small with respect to the radius , the R–S formula and numerical calculation are used to determine the Fresnel number. In laser systems, is usually much larger than ; therefore the expression is generally applicable.
4. Further improvement
The correction factors and are introduced into the definition of the Fresnel number in Equation (
Figure 7.Diagram of an obliquely incident spherical wave.
To further explore the correction factor under oblique incidence, an obliquely incident spherical wave is selected with a curvature center at point and a curvature radius of (Figure
Equation (
The off-axis Fresnel number is also applicable for non-circular apertures and complex optical systems. For a non-circular aperture, is independent of the incident angle; this value is a function of the aperture shape. Hence, is similar to the correction factor under normal incidence[
5. Conclusions
The off-axis Fresnel number is defined in this paper: . Theoretical analysis and numerical calculation yield ; is independent of the incident angle, but dependent on the incident wavefront and the aperture shape. In summary, the following conclusions are reached. (1) During propagation, normal and oblique incidences are consistent and can be unified by the axis transformation factor . (2) The Fresnel number can be utilized to prove that the effects of the incident wavefront and the aperture shape are not related to the incident angle . (3) The Fresnel number has a significant function in the design of laser systems, such as the effect of off-axis optical systems. Change of by oblique incidence or tilted optical elements improves the radial intensity distribution. This improvement reduces harm to optical elements, which is caused by diffraction-generated vibrations.
The Fresnel number is no longer confined to normal incidence; the number can be used to explain more diffractive optical phenomena, such as the design of off-axis laser systems. The expression is also suitable for incident waves of arbitrary shape, non-circular diffractive apertures and complex optical systems.
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